2 families of examples that are sometimes useful to have in mind:
(1) The group ring of a non-abelian finite group over a finite commutative ring.
and
(2) the incidence algebra of a finite poset over a finite commutative ring (the ring of upper triangular matrices is a basic example of this).
Of course, both of these are special cases of the same more general categorical (or quiver) definition. Before I wrote that I'd never dealt with the more general concept, but that was a lie...
Marc Wambst constructed a resolution for your algebra $A$, which he calls naturally enough a quantum Koszul complex. This is a projective resolution of $A$ as an $A$-bimodule and it looks like the usual bimodule Koszul resolution of polynomial rings sprinkled with $q$'s all over the place.
His quantum Koszul complex has length $n$ showing that $\operatorname{pdim}\_{A^e}A\leq n$. Now, a little homological algebra shows that $\operatorname{gldim}A\leq\operatorname{pdim}\_{A^e}A$ (this is explained in Cartan-Eilenberg, in the chapter on augmented rings), so now we know that the global dimension of $A$ is at most $n$. Computing $\operatorname{Tor}_A^n(k,k)$ is easy using his complex, and it is non-zero: this shows that we in fact have an equality, and
$$\operatorname{gldim}A=n.$$
As for your general question, there is no general method of computing the global dimension of rings. You will find several useful techniques which apply to interesting classes of algebras in the books by McConnell and Robson, and by Lam, among several other places —in general, it is somewhat of an art.
The paper I referred to above is [M. Wambst, Complexes de Koszul quantiques. Ann. Inst. Fourier (Grenoble) 43, 3 (1993), 1089–1159]. Every single time I have needed the explicit form of this resolution, though, I have found it easier to consruct it by hand :-)
Best Answer
At arsmath's request, I'm making this official. (This is pretty standard commutative algebra, but I realize not everyone has gone through it.)
A commutative ring $R$ is regular if it's noetherian and its local rings are regular. Using Serre's theorem e.g. Matsumura Commutative Ring Theory p 156, and the fact that $Ext$ commutes with localization, we can see that any regular ring with finite Krull dimension has finite global dimension.
To an algebraic geometer regular = nonsingular. So in particular, so there is a large supply of basic examples arising as coordinate rings of nonsingular affine varieties. This is a bit circular the way I'm saying it, but of course, you can test the condition using the Jacobian criterion...