universal enveloping algebra
A Lie bracket is not an associative product, but it badly wants to be a commutator inside some associative algebra. The universal enveloping algebra is the most economical associative algebra that grants this wish: it is built so that the abstract bracket [x, y] literally becomes the commutator xy - yx, and it does so as freely as possible, imposing no relations beyond the ones the bracket demands. It is the device that translates Lie theory into associative-algebra and module language.
Concretely, U(L) is the quotient of the tensor algebra T(L) by the two-sided ideal generated by all elements x (tensor) y - y (tensor) x - [x, y]. It comes with a Lie algebra map i : L -> U(L) and satisfies a universal property: any Lie algebra homomorphism from L into (the commutator Lie algebra of) an associative algebra A factors uniquely through an associative algebra homomorphism U(L) -> A. In particular, representations of L are exactly the same as modules over U(L).
The Poincare-Birkhoff-Witt (PBW) theorem pins down the size of U(L): if x_1, ..., x_n is an ordered basis of L, then the ordered monomials x_1^{a_1} ... x_n^{a_n} form a basis of U(L). In particular the map i : L -> U(L) is injective, so L really does embed in an associative algebra, and U(L) is as large as a polynomial ring even though L itself may be highly noncommutative.
For the 1-dimensional abelian Lie algebra L = span(x), there are no nontrivial brackets, so U(L) = k[x], the polynomial ring in one variable. For the Heisenberg algebra with [x, y] = z (z central), U(L) is the Weyl-algebra-like ring where xy - yx = z holds as an honest identity.
U(L) of an abelian L is a polynomial ring; of the Heisenberg algebra, a Weyl-algebra relative.
When L is abelian the relations x (tensor) y - y (tensor) x - 0 just impose commutativity, so U(L) is the polynomial ring (symmetric algebra) on L. PBW says that in general U(L) is, as a vector space, no bigger than this commutative shadow.