A differential graded algebra (dg-algebra) is a monoid object in the category of chain complexes with respect to the usual tensor product of complexes. A (graded) commutative dg-algebra is simply a commutative monoid object.
The de Rham complex for a smooth manifold is easily seen to be a commutative dg-algebra with respect to the wedge product.
I could have sworn that I saw somewhere that the de Rham complex is the free dg-algebra (or free commutative dg-algebra?) on some appropriate base category, i.e. left adjoint to a forgetful functor. I can't seem to find a reference for this, however.
With regards to my background I'm familiar with category theory and differential geometry, but my algebra is a bit weak.