In the non-commutative situation, it is called Groebner-Shirshov basis. See, for example, <a href="http://arxiv.org/abs/0804.1344"> this </a> survey paper and references there. The case of universal enveloping algebra was considered by Bergman in G. M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29, 178-218(1978). I do not know about representation theory, but certainly this is one of the main tools of studying finitely presented rings and algebras over a field. The problem, of course, is that a finite Groebner-Shirshov basis (unlike Groebner basis in the commutative case) does not always exist.