Alain's answer is the best application I heard about. I cannot add something comparable. Just two "motivations" which are somewhat nice to me. However they does not answer yours questions, sorry.
General claim - studying non-commutative objects is useful for understanding commutative ones.
Subclaim - non-commutative algebras can be equivalent (Morita, Koszul dual or whatever) to non-commutative ones, however non-commutative "models" can provide more easy way to study commutative things.
Examples 1. Consider commutative algebra A of functions on manifold M and group G. You may be interested in factor $M/G$ which is related to invariants $A^G$.
Claim. Under certain conditions COMMUTATIVE $A^G$ is Morita equivalent to NON-COMMUTATIVE $A\cross A\times C[G]$ - cross-product algebra of $A$ and group algebra of $G$. In some cases it is more easy to work with this cross-product sometimes it can be described more explicitly. You may see just the first sentences in Etingof Ginzburg famous paper: http://arxiv.org/abs/math/0011114
Example 2. Quantization. Our real world is actually quantum. So physicists are interested in this. Mathematical way to understand quantization is a procedure to construct the non-commutative algebras from commutative ones. The big mathematical challenge is to understand how to relate properties on non-commutative quantum algebras to properties of commutative ones. Probably the most striking and most simple formulated is the conjecture that automorphisms group of classical symplectic R^2n and quantum (i.e. just the algebra of differential operators in n-variables) are isomorphic. http://arxiv.org/abs/math/0512169 Automorphisms of the Weyl algebra Alexei Belov-Kanel, Maxim Kontsevich
It is somewhat related to the famous Jacobian conjecture. See http://arxiv.org/abs/math/0512171