A Weyl algebra is the noncommutative algebra of algebraic differential operators on a polynomial ring. To each variable
x corresponds the operator
dx that differentiates with respect to that variable. The evident commutation relation takes the form
dx*x == x*dx + 1.
We can give any names we like to the variables in a Weyl algebra, provided we specify the correspondence between the variables and the derivatives, with the
WeylAlgebra option, as follows.
i1 : R = QQ[x,y,dx,dy,t,WeylAlgebra => {x=>dx, y=>dy}]
o1 = R
o1 : PolynomialRing, 2 differential variable(s)

i2 : dx*dy*x*y
o2 = x*y*dx*dy + x*dx + y*dy + 1
o2 : R

i3 : dx*x^5
5 4
o3 = x dx + 5x
o3 : R

All modules over Weyl algebras are, in Macaulay2, right modules. This means that multiplication of matrices is from the opposite side:
i4 : dx*x
o4 = x*dx + 1
o4 : R

i5 : matrix{{dx}} * matrix{{x}}
o5 =  xdx 
1 1
o5 : Matrix R < R

All Gröbner basis and related computations work over this ring. For an extensive collection of Dmodule routines (A Dmodule is a module over a Weyl algebra), see Dmodules.
The function isWeylAlgebra can be used to determine whether a polynomial ring has been constructed as a Weyl algebra.
i6 : isWeylAlgebra R
o6 = true

i7 : S = QQ[x,y]
o7 = S
o7 : PolynomialRing

i8 : isWeylAlgebra S
o8 = false
