In my view Hochschild cohomology is the most interesting cohomology on associative (and I dare say commutative algebras). So far all that has been said is about different methods of computation. But there are also many applications and ways of viewing it.
Skip the following paragraph if you want, it's just a side point.
The one that sticks in my mind is the
application to deformation theory.
The Hochschild cochain complex is
actually the object of interest in
deformation theory, its homology is
just one invariant of it and captures
the infinitesimal deformations. The
cochain complex carries a specific
algebraic structure; it's an algebra
for the braces operad. And then
there's the celebrated (and many times
proved ;-)) Deligne conjecture which
says that it may be viewed as a
homotopy Gerstenhaber algebra.
Finally there's Kontsevich's formality
result which says that for smooth
commutative algebras that looking at
homology and its Gerstenhaber algebra
structure actually does capture all
information about the Hochschild
cochains and hence the deformation
theory of the algebra.
Anyway I didn't mean the write that, but just got overexcited, my point in writing this answer was to say that there are other homology theories.
For example there's the bar homology. This homology is little known which is a big pity because it's actually rather special! There's a very good reason why it's not studied though and that's because for a unital algebra its homology is always zero, but it is still interesting because it the chain complex a coalgebra and we're not interested in its homotopy type as a complex and so shouldn't be taking its homology at all! The coalgebra actually gives generators and relations for the algebra, it's the derived functor of
$A \mapsto A/(A.A)$
from the category of associative algebras to vector spaces.
But you guys like taking homology, so I should give you a better reason for studying the bar homology. Suppose you have an augmented algebra, so we can split the identity off and write
$A = k\oplus A'$
Then the bar homology of $A'$ is not necessarily zero and gives interesting invariants of the algebra. In the char 0 commutative case this is well studied, you guys might know it as part of rational homotopy theory. The commutative bar homology of the cohomology ring of a nice space is the rational homotopy of the space.