Why is it useful to study vector bundles? I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just wondering, is there any big reason why the study of bundles would give information about varieties? (I suppose that for this matter, I should actually replace varieties by manifolds?)
I have heard about some invariants, like the Picard group for complex manifolds. But given my inexperience in these concepts, I don't really know why they should be important. So for those who are thinking about "cooking up invariants", some more detailed explanations on why they are useful (and hopefully some elementary examples!) would be appreciated. Thanks!
 A: Here are some motivations from the point of view geometric quantization. In this theory, one usually considers vector bundles over a symplectic manifold. The symplectic manifold represents the state space of a "classical mechanical system", for example in the case of a particle moving on a line, the symplectic manifold is $\mathbb{R}^2$, the space of the particle's  positions and momenta. 
The equations of motion in classical mechanics (Hamilton's equations) are in general nonlinear in the state space coordinates, since they are obtained from Poisson brackets. The quantum version of this problem considers line bundles over the "classical" symplectic space. The sections of these bundles represent the particle's "wave functions". In quantum mechanics, the equations of motions are linear (The Schrodinger equation). This "linearization" is achieved by working on the line bundle which has an intrinsically linear structure, and the induced evolution operators act linearly on the space of sections.
Many properties of line bundles have significance here also. For example, the projective space into which the sections of the line bundle define an embedding is just the projective quantum mechanical Hilbert space. 
The generalization to a vector bundle is used to describe particles with "internal degrees of freedom" such as spin.
Harmonic analysis on vector bundles is used in spectrum problems of the quantum mechanical case.
A: They are tractable and naturally occuring yet encode lots of information. They also provide a link between different mathematical techniques. One good comparison is between solving Yang Mills equations on a vector bundle and the Einstein equations on a Riemannian manifold.
K-Theory shows that there is lots of topological information contained in them.
Hitchin Kobayashi correspondence linking differential and algebraic techniques, the Atiyah Singer index theorem linking analysis and topology. Flat connections and curvature link geometry and representation theory. The Torelli theorem and Donaldson's work use them to reveal information about finer structures (algebraic and differentiable respectively). They occur naturally, tangent and normal bundles obviously but also projective embeddings.
Hitchin's paper The Self Duality Equations on a Riemann Surface combines many of them beautifully.
A: Following Gowers' answer (which is what I would like to have written) and Siu's response, it seems to me that a common theme in mathematical research of the past, say, 50 years is something like the following:
1) Extract from examples the definition of an abstract mathematical object (Riemann surfaces, algebraic varieties, vector bundles, etc.)
2) Define the set or space of all such abstract objects and look for some kind of structure, usually algebraic or topological, that exists naturally on this space. If necessary, impose an equivalence condition (homeomorphic, homotopic, homologous)
3) By analyzing and classifying such spaces (which may depend on parameters), develop novel insights into the original examples that originally led to the abstract definition.
So, returning to vector bundles, once you have a theory for classifying and distinguishing different types of vector bundles, you can apply it to naturally defined bundles over manifolds or varieties and obtain new results on classifying and distinguishing different types of such spaces.
Vector bundles are particularly attractive, because they represent "linearizations" of the nonlinear structure of manifolds and varieties. So they are in many ways much easier to work with than the base spaces.
A: Well, in algebraic geometry, here's a couple of reasons:
1) Subvarieties: Take a vector bundle, look at a section, where is it zero? Lots of subvarieties show up this way (not all, see this question) but generally, we can get lots of information out of vector bundles regarding subvarieties.
2) Invariants of spaces: The Picard group of Line bundles and more generally the Grothendieck group/ring is a useful invariant for differentiating spaces and analyzing the geometry indirectly.  On smooth spaces, in fact, complexes of vector bundles can be used to replace coherent sheaves entirely (I believe by the Syzygy Theorem).
3) Maps into Projective Space: This one is line bundle specific.  Let $V\to\mathbb{P}^n$ be any imbedding, say, then the pullback of $\mathcal{O}(1)$ is a line bundle on $V$.  The nice thing is, the global sections of this line bundle determine and are determined by the map (we can get degenerate mappings by taking subspaces, but lets ignore that, and base loci for the moment).  It turns out that we can define a line bundle to be ample, a condition just on the bundle, and that suffices to say that a power of it gives a morphism to $\mathbb{P}^n$, so understanding maps into projective space is the same thing as studying ample line bundles on a variety.
Hope that helps, there's a lot more, but those are the first three things that came to mind.
A: It is very useful in several areas of mathematics to study vector bundles, of which tangent bundles are a particular case for the following 4 reasons.

*

*Vector Bundles give rise to interesting examples: first, the trivial bundle, $V= X \times \mathbb{R}^r$ where $X$ is a topological space and we're looking at real vector spaces of dimension $r$ here. For each point $x\in X$ we define $V_x= x \times \mathbb{R}^r$. We can also study the mobius band using vector bundles: this is part of the larger idea that vector bundles are used to classify surfaces and detect to which degree they are contradictible (fourth point below). Vector bundles are important in studying finitely generated projective modules in algebraic geometry: projective modules corespond to rank-1 vector bundles on the algebraic variety we're looking at, or line bundles.


*Given a vector bundle $V$, we can talk about the section of the vector bundle $V$. These sections give rises to generalization of vector valued functions on the topological space $X$. So in particular, if we consider tangent bundles, the sections will be vector fields. In algebraic geometry there are very few globally defined functions so we need this more general notion is algebraic geometry. The sections figure in mappings of projective space for examples: they allow you to map $\mathbb{C}\mathbb{P}^1$ to $\mathbb{C}\mathbb{P}^2$ in a canonical fashion.


*The third reason why studying vector bundles is important is when studying 2 topological spaces $X$ and $Y$ and $X$ is immersed in $Y$. Say they're manifolds. The normal bundle $N$ on $X$ defined by $N_x=T_x Y / T_x X$ for each $x\in X$. The zero section given by $\pi: X\to N$ is a first order approximation of how $X$ sits in $Y$: $\rho: X\to Y$, so this allows you to study submanifold of manifolds by recording the infinitesimal information about how $X$ sits inside $Y$.


*Vector bundles on $X$ give us a way to study $X$. Consider the following theorem: if $X$ is compact, Haussdorf and contractible then every vector bundle over $X$ is trivial. Therefore, collections of isomorphism classes of vector bundles on $X$ measure how far from being contractible $X$ can be. This gives us an invariant with which to study a topological space $X$. This leads to the study of $K$-theory, an important area of mathematics.
A: I think many of the other answers boil down to the same underlying idea: Sections of vector bundles are "generalized functions" or "twisted functions" on your manifold/variety/whatever.
For example, Charles mentions subvarieties, which are roughly "zero loci of functions". However, there are no non-constant holomorphic global functions on, say, a projective variety. So how can we talk about subvarieties of a projective variety? Well, we do have non-constant holomorphic functions locally, so we can still define subvarieties locally as being zero loci of functions. But the functions $f_i$ which define a subvariety on one open set $U$ and the functions $g_i$ which define a subvariety on another open set $V$ won't necessarily agree on $U \cap V$. We need some kind of "twist" to make the $f_i$'s and the $g_i$'s match up on $U \cap V$. Upon doing so, the global object that we obtain is not a global function (because, again, there are no non-constant global functions) but a "twisted" global function, in other words a section of a vector bundle whose transition functions are described by these "twists".
Similarly, sections of vector bundles and line bundles are a nice way to talk about functions with poles. Meromorphic functions then become simply sections of a line bundle, which is nice because it allows us to avoid having to talk about $\infty$. This is essentially why line bundles are related to maps to projective space $X \to \mathbb{P}^n$; intuitively, $n+1$ sections of a line bundle over $X$ is the same as $n+1$ meromorphic functions on $X$, which is the same as a map "$X \to (\mathbb{C} \cup \infty)^{n+1}$" which becomes a map "$X \to \mathbb{P}^n$" after we "projectivize".
One way to think of vector bundles and their sections as being invariants of your manifold/variety/whatever is to think of them as describing what kinds of "generalized" or "twisted" functions are possible on your manifold/variety. 
The view of sections of vector bundles as being "twisted functions" is also useful for physics, as in e.g. David's answer. For instance, suppose we have a manifold, which we think of as being some space in which particles are moving around. We have local coordinates on the manifold, which are used to describe the position of the particles. Since we are on a manifold, the transitions between the local coordinates are nontrivial. We may also be interested in studying, say, the velocities or momenta (or acceleration, etc.) of the particles moving around in space. On local charts we can describe these momenta easily in terms of the local coordinates, but then for a global description we need transitions between these local descriptions of momenta, just like how we need transitions between the local coordinates in order to describe the manifold globally. The transitions between local descriptions of momenta are not the same as that between the local coordinates (though the former depends on the latter); phrased differently, we obtain a non-trivial (ok, not always non-trivial, but usually non-trivial) vector bundle over our manifold.
A: Although not a complete answer to the question, let me just point out that vector bundles are sometimes forced upon you.
For instance, you may start with an honest function $f$ defined on a manifold $M$ on which a group $G$ acts.  Let's assume for simplicity that $G$ acts in such a way that the quotient $M/G$ is a manifold.  If the function were invariant under the group, it would define an honest function on the quotient.  But if the function is "almost" invariant, say
$$f(g^{-1} x) = \alpha(g) f(x)$$
for $g\in G$ and $x \in M$ and where $\alpha$ is some character of $G$, then $f$ only defines a section of a (homogeneous) line bundle on the quotient.
More generally if $f: M \to V$, where $\rho: G \to \mathrm{GL}(V)$ is a representation of $G$, and assuming that
$$f(g^{-1} x) = \rho(g) f(x)$$
then in the quotient $M/G$, $f$ defines a section of a (homogeneous) vector bundle.
Another case is where you have a family of endomorphism $\phi(x) \in \mathrm{End}(V)$ of a fixed vector space $V$, parametrised by a manifold $M$.  Then the kernel of $\phi(x)$ is a vector subspace of $V$, and assuming that its dimension does not vary with $x$, define a vector bundle over $M$.
Also there are interesting invariants which require one to consider vector bundles.  For instance, topological K-theory, which is the natural setting for the index theorem, is a theory of vector bundles.
Finally, vector bundles are essential for gauge theory which in turn have provided very useful results in topology: Donaldson's early work in the 80s on the topology of 4-manifolds, Seiberg-Witten theory in the mid 1990s,...
A: in a very precise sense, vector-bundles ARE 'representations' of the base geometric object M, i.e. topological space, or a topological manifold, differentiable manifold, a complex manifold, analytic variety, Algebraic variety, etc, each of which is characterized by an  algebra of appropriate type of functions. Then representations of this algebra are projective modules, each of which is isomorphic to module of sections of some vector hundle. (topological, smooth, algebraic, analytic, etc...) So, then the ring of such vector bundles under whitney sum and tensor product is a very important invariant of M.  On the other hand, for indívidual specifíc vector bundles, the modules of sections arise natirally in geometric problems---eg, derivatives of functions in smooth case or study or differential geometry  of a riemannian structure, or maps of an algebraic variety into projective spaces---in short a vector bundle represents some aspect of the geometry, and the collection of v.b's represent the more primary geometric structure of the base...
A: Euclid's geometry is the geometry of frozen objects (hence Russell's description of mathematics as "supreme beauty—a beauty cold and austere"). Differential calculus is Euclidean geometry that moves, i.e. geometric objects with rates of change. Any moving subvariety $X\subset Y$ [map $\varphi \colon X \to Y$] has derivative in $\varphi^*TY/TX$ [a section of $T^*X \otimes \varphi^*TY$], a vector bundle [section of vector bundle]. So the first order deformations of nonlinear objects (maps, etc.) are vector subbundles [sections of vector bundles]. Higher order deformations similarly. So vector bundles are where we do calculus, make linear approximations, when we move things (maps, subvarieties).
A: Another example where bundles are forced on you is if you want to differentiate a function on a manifold. (This may be a special case of one of the remarks above -- I write as a total non-expert.) If you differentiate a real-valued function on R^n you get a function that takes values in R^n: if you differentiate a real-valued function on an n-dimensional manifold then it takes values in the tangent bundle. This doesn't really explain why they are such a powerful concept, but at least it shows that they are a natural one.
A: In Mirror Symmetry, specially in Homological Mirror Symmetry, the mirror of stable holomorphic Vector bundle is the special Lagrangian submanifold.
In symplectic geometry
A-brane = Lagrangian submanifold + flat vector bundle •
In holomorphic geometry
B-brane = complex submanifold + holomorphic bundle
There is a relation between reflexive Sheaf and Vector bundle which is used in the study of Extension theory for finding canonical metrics
A reflexive sheaf $ F$, on Kähler variety $X$ outside of codimension at least 3 is a holomorphic vector bundle,
In the study of positivity theory of direct image of Line bundles which are Vector bundles.
For example positivity of CM-bundle which is related to K-stability and Kähler-Einstein metric on Fano manifolds
A: Calculus is extended from vector spaces to manifolds by way of the tangent bundle. This is a very specific kind of vector bundle. The Cartan calculus of differential forms is built over it as well as the older Ricci calculus so beloved by physicists. This already proves that vector bundles are valuable. But further, as Einstein taught us through his notion of general covariance, looking at object including its isomorphisms is valuable, hence we should not look at just tangent bundles but also all vector bundles that are isomorphic to it. Such bundles are used in the gauge interpretation of GR used, say, in Loop Quantum Gravity.
