It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.
Exterior algebras are very useful algebraic objects. Exterior powers are extensively used in differential geometry. Questions related to plane fields involve the exterior powers as they can be realised as the kernel of a differential form, locally. The exterior algebra has another vital thing about it, that is exterior differentiation. Also, a direct application is Stokes' theorem which is used in measuring areas, volumes, performing integration on surfaces.