Given a field $k$, the $n$-th Weyl algebra $A_{n,k}$ is an associative unital algebra over $k$ generated by the symbols $x^1,\ldots,x^n,\partial_1,\ldots,\partial_n$ modulo relations $x^i x^j = x^j x^i$, $\partial_i\partial_j = \partial_j\partial_i$ and $\partial_i x^j - x^j \partial_i = \delta_i^j$ (the Kronecker delta).
In characteristic zero, it agrees with the algebra of regular differential operators on the $n$-dimensional affine space.
Sometimes one considers the Weyl algebras over an arbitrary $k$-algebra $R$, including noncommutative $R$, when the definition is simply $A_{n,k}\otimes_k R$. Another generalization are the symplectic Weyl algebras.
In quantum physics, one often studies Weyl algebras over the complex numbers; the usual notation there is $p_j$ for $- \mathrm{i} \partial_j$ (where $\mathrm{i}$ is the imaginary unit).
Please distinguish from Weil algebra.
The Weyl algebra on $2n$ generators is the quotient of the universal enveloping algebra of the Heisenberg Lie algebra on $2n$ generators, obtained by identifying the central elements of the Heisenberg Lie algebra with multiples of the identity element.
S. C. Coutinho, A primer of algebraic D-modules, London Math. Soc. Stud. Texts 33, Cambridge University Press 1995. xii+207 pp.
eom: J.-E. Björk, Weyl algebra
A categorification of the Weyl algebra is introduced in operadic language in