The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential geometry course (I was the Reeb foliation) and I didin't pay many attention to it. In the meanwhile I get interested in the noncommutative theory in particular in $C^*$algebras. While reading about Noncommutative Geometry I came across foliations as the one of the main motivating examples of the theory. I learned that in general the space of leaves of the foliation is badly behaved as a topological space and I believe that it is more worthwile to deal with these spaces using algebraic methods. But I don't have something like a mental picture of what foliations is and why should I even care about those objects?

5$\begingroup$ Leaf spaces of a foliations are examples of stacks, and there's no need to go to C*algebras to talk about them. Now... you might what to ask another question: "What is a stack and why should I care?". ;) $\endgroup$ – André Henriques Feb 7 '16 at 1:18

4$\begingroup$ On a philosophical note, Connes motivates foliations as the simplest geometry of von neumann factorization and von neuman factorization as how time emerges. (Relevant excerpt from Connes' (english) Temps et Aléa du Quantique (youtube.com/watch?v=ODAngTW8deg) talk excerpted here: ncatlab.org/nlab/show/time). $\endgroup$ – Trent Feb 7 '16 at 4:59

4$\begingroup$ @truebaran The holonomy groupoid of a foliation is used to associate a C* algebra to a foliation. This groupoid is based on the concept of holonomy of a leaf. The later is a generalization of the Poincare return map of a closed orbit of a vector field.BTW the NCG methods can be used to prove that the Kronecker foliation with slops $\alpha$ and $\beta$ are not topological equivalent if these slops are not in the same orbit of the action of $Sl(2,\mathbb{Z})$. $\endgroup$ – Ali Taghavi Feb 7 '16 at 9:09

2$\begingroup$ This is a paper on NCG aspects of foliation sciencedirect.com/science/article/pii/0022123685900382 $\endgroup$ – Ali Taghavi Feb 7 '16 at 9:11

5$\begingroup$ In general relativity you see foliations over and over as ways of discussing Cauchy problems and the time evolution of systems. These are foliations using spacelike surfaces. Because GR doesn't have a preferred time coordinate or a preferred notion of simultaneity, a foliation in spacelike surfaces is the next best thing. $\endgroup$ – Ben Crowell Feb 7 '16 at 15:11
Without any disrespect, let me say that I find it incredible that someone naturally cares about noncommutative geometry but needs convincing about actual geometry (this just goes to highlight that there is a wide variety of ways of thinking in mathematics). I would need convincing the other way around (e.g. How are C* algebras relevant in foliation theory from the geometric point of view?).
From the point of view of someone interested in geometry, foliations appear naturally in many ways.
The most basic way is when you consider the level sets of a function. If the function is a submersion you get a nonsingular foliation, but this is rare. However every manifold admits a Morse function and the theory of Morse functions (which can be used for example to classify surfaces, and to prove the high dimensional case of the generalized Poincaré conjecture) can be seen as a special (or maybe as the most important) case of the theory of singular foliations (where the singularities are pretty simple).
Another natural type of foliation is the partition of a manifold into the orbits of the flow determined by a vector field. Again the simplest case, in which the vector field has no zeros, is rare but yields a nonsingular foliation (with onedimensional leaves). However, already in this case one can see that the leaves of a foliation can be recurrent (i.e. accumulate on themselves) in nontrivial ways (the typical example is the partition of the flat torus $\mathbb{R}^2/\mathbb{Z}^2$ into lines of a given irrational slope).
A notable fact generalizing the above case (the result is in papers of Sussmann and Stefan from the early 70s) is the following: Consider $n$ vector fields on a manifold. For each point $x$, consider the set of points you can reach using arbitrary finite compositions of the flows of these vector fields. The partition of the manifold into these "accessibility classes" is a singular foliation (in particular each accessibility class is a submanifold).
Hence foliations appear naturally in several types of "control problems" where one has several valid directions of movement and wishes to understand what states are achievable from a given state. This point of view also gives a nice insight into Hörmander's theorem on why certain differential operators have smooth kernels (there's a nice article by Hairer explaining Malliavin's proof of this theorem). Essentially the Hormander bracket condition means that Brownian motion can go anywhere it wants (i.e. a certain foliation associated to the operator is trivial).
Another way to obtain motivation is to look at history (I remember reading a nice survey which I think was written by Haefliger). In my (unreliable) view, the first geometric results (so I'm skipping Frobenius's theorem) in foliation theory are the PoincaréBenedixon and PoincaréHopf theorems both of which can be used to show that every onedimensional foliation of the twodimensional sphere has singularities.
Hopf then asked in the 1930's if there exists a foliation of the three dimensional sphere using only surfaces. The first observation, due to Reeb and Ehresman is that if one of the surfaces is a sphere then you cannot complete the foliation without singularities. They also constructed the famous Reeb foliation and answered the question in the affirmative.
Since then there has been a whole line of research dedicated to the question of which manifolds admit nonsingular foliations. In this regard, the main Theorem is due to Thurston who (in the words of an expert in the theory) came around and "foliated everything that could be foliated".
But there are other lines of research. For example, I know that there is a certain subset of algebraic geometry dedicated to trying to understand the foliations of complex projective space which are determined by the level sets of rational functions of a certain degree.
Also, whenever you have an action of the fundamental group of a manifold there is a natural "suspension" foliation attached (suspensions are considered the "local model" for a general foliation and are hence very important in the theory). This point of view sometimes has given results in the current area of research known as higherTeichmüller theory (where basically they study linear actions of the fundamental group of a surface).
And of course, when one has an Anosov, or hyperbolic, diffeomorphism or flow (for example the geodesic flow of a hyperbolic surface), there are the stable and unstable foliations which play a role for example in the famous Hopf (not the same Hopf as before) argument for establishing ergodicity.
Oh, and I haven't even mentioned the special place that foliations occupy in the theory of 3dimensional manifolds. Here there are many results which I can't say much about (but I've heard the book by Calegari is quite nice). Maybe a basic one is Novikov's theorem which basically proves that the existence of Reeb components is forced for foliations on many 3manifolds.
And (I couldn't resist adding one last example), there are also foliations by Brouwer lines, which have recently been used (by LeCalvez and others) to prove interesting results about the dynamics of surface homeomorphisms.
TLDR: Foliations occur naturally in many contexts in geometry and dynamical systems. There may not be a very unified "Theory of foliations" but several special types have been studied in depth for different reasons and have yielded insight or participate in the proof of important results such as the Poincaré conjecture and Hörmander's bracket theorem. For this reason mathematicians have taken notice and singled out foliations as a basic object in geometry (there have even been significant efforts in producing a couple of nice treaties trying to give the grand tour, for example the books by Candel and Conlon).

15$\begingroup$ Speaking as a functional analyst, +1 for the first sentence :) $\endgroup$ – Yemon Choi Feb 7 '16 at 15:27


4$\begingroup$ Honestly the first sentence is pretty needlessly provocative, and is near offensive when you generalise it to all of mathematics. It's akin to saying "I find it hard to believe that anyone is interested in pure mathematics and not applied mathematics". $\endgroup$ – Miles Rout Feb 10 '16 at 6:06

20$\begingroup$ @Miles Rout: Well that's not the way I intended it. And, considering the comments above yours, not everyone interpreted it that way either. I believe the diversity of motivations and mindsets is a positive thing. $\endgroup$ – Pablo Lessa Feb 10 '16 at 19:51

$\begingroup$ The phrasing is a bit careless, but I think perhaps it can be forgiven a bit as English is not the first language of the OP. $\endgroup$ – Hollis Williams Aug 16 '20 at 12:56
Probably there are many reasons why people care about foliations, but for someone coming from operator algebras one of the main reasons is the connection to von Neumann algebra theory. In brief, every foliation of a smooth manifold has an associated von Neumann algebra and interesting properties of the von Neumann algebra are reflected in geometric properties of the foliation. The von Neumann algebra is a factor if and only if the foliation is ergodic, for example. You can get examples of factors of all types by this construction and the geometric aspect of foliations is especially helpful in understanding the type III case. The modular automorphism group has a straightforward geometric interpretation, and so on. A good reference is "Operator algebras and the index theorem on foliated manifolds" by H. Moriyoshi.
Here is another reason about why people care about foliations: If you care about dynamical systems, you should care about foliations.
For instance, if you have a nowhere singular vector field on a closed manifold, it defines a 1dimensional foliation, the leaves are the orbits of the flow associated to the vector field. Studying 1dimensional foliations is (almost, modulo orientation issues) the same as studying nonsingular vector fields. The only thing is that when you think in terms of foliations, you forget the parametrization of the orbits.
If you have a locally free action of a connected Lie group on a closed manifold, again you have an associated foliation by the orbits. Here locally free means that the stabilizer of any point in the manifold is a discrete subgroup of the Lie group. The dimension of the leaves will be the dimension of the Lie group, this generalizes example 1. There are already interesting things going on for actions of the Lie group $R^2$ (i.e., pairs of commuting vector fields).
Even if you are only interested in flows or diffeomorphisms, there are sometimes natural foliations associated to them. For instance, the stable/unstable foliations of an Anosov flow or diffeomorphism.
As for the mental picture of what a foliation is, well there are probably plenty of pictures online. You should think of a partition of your manifold which is locally nice (like a "millefeuille" or like a product $R^{k}\times R^{nk}$) but globally complicated (leaves could be dense or their closure could be transversally a Cantor set, etc.).
Here's why I care about foliations. It is always interesting when a structure can be expressed in terms of simpler structures. For instance a torus is the union of circles making it into a cartesian product. Or a Klein bottle is the union of circles making it into a nontrivial fibre bundle.
A foliation is a description of an nmanifold M as the union of submanifolds of fixed lower dimension k, whose tangent planes fit together cleanly. (I.e., such that every point of M has a neighborhood that is decomposed topologically just as ndimensional Euclidean space is the cartesian product of kdimensional Euclidean space and (nk)dimensional Euclidean space.) This is a natural generalization of two things: a) the concept of a fibre bundle, and b) the decomposition of a manifold into the trajectories of a nonsingular vector field.
A foliation can be defined in terms of the reduction of a manifold's atlas to a certain simple pseudogroup.
The quintessential example of a foliation is the Reeb foliation of the 3sphere. Although the 3sphere possesses infinitely differentiable codimension1 foliations such as the Reeb foliation, it is a beautifully subtle fact, proved by André Haefliger, that it has no realanalytic one. Another fascinating fact, proved by Sergei Novikov, is that every codimension1 foliation of the 3sphere must possess a compact leaf (which is in fact a torus). Yet another fascinating fact, proved by Krystyna Kuperberg, is that the 3sphere possesses a realanalytic 1foliation that has no closed trajectory. (This counterexample was discovered only 43 years after the original conjecture, by Herbert Seifert, suggested that every 1foliation of the 3sphere must have a closed trajectory.)