Different definitions of the dimension of an algebra I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:


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*The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.

*The Krull dimension, based on chains of prime ideals.

*The transcendence degree of the fraction field of A over F.


According to Artin, GK dimension is the most robust notion because it applies to certain noncommutative algebras.  And in the noncommutative setting one can't form fraction fields, so the transcendence degree is out of the question (right?).
But Krull dimension has the advantage that it applies to arbitrary rings.  As far as I can tell, the definition of GK dimension only applies to algebras.  So: as a matter of pedagogy, which notion of dimension is most appropriate for what applications?  Which is easiest to prove things about when?
 A: Often the most useful dimension in non-commutative algebra is the length of the minimal injective resolution of the ring as a module over itself. In many important cases this is the same as the global dimension when the latter is finite, but it is more robust in that it is finite more often. In a commutative Noetherian ring it is the same as the Krull dimension when it is finite.
A: I agree with Greg Muller that homological dimension is very nice. In fact various flavours of it turn out to be the same as Krull dimension. For instance a local commutative ring with unit is regular if and only if it has finite global dimension and in this case the global dimension is precisely the Krull dimension.  
If there are singularities present so that the global dimension is infinite one can then turn to relative homological dimensions. For instance the right global Gorenstein injective dimension of an n-Gorenstein (not necessarily commutative) ring is n and in fact one can detect Cohen-Macaulayness using a criterion involving Gorenstein injective dimension. One can extend these notions to (at least certain) abelian categories.
To actually try and answer some of your question now - at least in commutative algebra I agree with Greg again that (locally) depth tends to be one of the most useful interpretations of the dimension in the cases (Cohen-Macaulay rings) when it agrees with dimension. Although as I sort of alluded to above one can prove results using homological notions of dimension as well. I don't really know the non-commutative theory well enough to comment in that case.
A: I know there has been no action on this question in awhile, but I was surprised no one mentioned weak dimension and I wanted to weigh in. I'm not trying to take anything away from right global dimension (it's my favorite too), but weak dimension seems to work in just as much generality as global dimension and may rule out some pathologies. For example, the right weak dimension (looking at flat dimension of right $R$-modules) is equal to the left weak dimension. Furthermore, weak dimension really does give different information from the notions of dimension discussed so far (for $R$ non-Noetherian). For example, rings of weak dimension zero are exactly Von Neumann Regular rings (these are also the rings for which all prime ideals are maximal).
An article you might enjoy which computes all the dimensions discussed so far for a particular ring is at this link: http://arxiv.org/abs/0912.0723
A: There's actually a Gelfand-Kirillov transcendence degree for a noncommutative algebra over a field that generalizes the classical transcendance degree.  Gelfand and Kirillov introduced it to prove that the field of fractions for the $n$th Weyl algebra is different for different $n$.  Here's a paper by James Zhang that gives the definition and calculates it for several examples.
There's actually a pretty nice theory for when you can form a field of fractions.  A Noetherian domain always has a field of fractions, in a construction that closely resembles the commutative case.  (This is a special case of Goldie's Theorem.)  For a non-Noetherian domains, things are more complicated: the domain can fail to have a field of fractions, and when it does, it won't usually have a unique smallest field it embeds into (unlike the commutative or noncommutative Noetherian case).  
A: In non-commutative algebra Krull dimension has been generalised by Gabriel & Rentschler. A decent account of it can be found in Chapter 6 of McConnell and Robson's book on non-commutative Noetherian rings. 
The basic idea is as follows: An artinian module has Krull dimension 0. 
A module that does not have Krull dimension 0 has Krull dimension 1 if in every infinite descending chain of submodules all but finitely many composition factors have Krull dimension 0.  
A module that does not have Krull dimension 0 or 1 has Krull dimension 2 if in every infinite descending chain of submodules all but finitely many composition factors have Krull dimension 0 or 1.
The definition continues for all finite ordinals (and can be extended to all ordinals). Then the Krull dimension of a ring R is the Krull dimension of R as a module over itself.
A: In a non-commutative ring, you need to be careful with what you even mean by a prime ideal, and usually there are very few two-sided ideals you might call prime.  Oh, and even in the cases when there is a nice ring of fractions, it won't be a field, and so transcedence degree is still bad.
My personal favorite notion of dimension is 'global dimension', the maximum projective dimension of any module of the ring.  This concept exists for any ring, and in fact for any abelian category (though, if there aren't enough projectives, you need to play with the definition).  The only problem is that it can often be infinity, even for relatively mild rings, like C[x]/x^2.  It still makes for a pretty good theory of 'smooth dimension', however.
From a conceptual perspective, Krull dimension seems best suited for geometric perspectives, since it is measuring chains of irreducible closed subsets.  The easiest times to work with Krull dimension is when you are in a Cohen-Macaulay ring, and then Krull dimension is equivalent to depth, which is easier to prove things about, since you only need to produce a maximal regular sequence.
A: Expanding the last entry on noncommutative transcendence degrees:
When your algebra $A$ is prime Goldie (such as Noetherian domains) there are two recent analogues of noncommutative transcendence degree, the lower transcendence degree (J. J. Zhang, On lower transcendence degree, MR1654189), and the homological transcendence degree (A. Yekutieli and J. J. Zhag, Homological transcendence degree, MR2235944), which are very usefeul in understanding the classical quotient ring $Q(A)$.
Lower transcendence degree is closely connected to Gelfand-Kirillov dimension and Gelfand-Kirillov transcendence degree; homological transcedence degree is connected to homological dimensions (building on previous work of Resco, Rosenberg, Schofield and Stafford).
Both invariants have interesting connections with difficult problems in ring theory and birational study of non-commutative projective schemes, have good theoretical properties (which the Gelfand-Kirillov transcendece degree lacks), and have been computed for a lot of division algebras.
