If you don't know what any of those words mean, then it's a way to represent n-dimensional geometric objects and operations on them all within an algebraically closed environment. Moreover, (at least in the specific algebras I'm familiar with) you can view a lot of transformations/operations as reflections or compositions of reflections of one object with respect to another.
It's quite beautiful really. Transformations compose nicely and produce new (uniquely represented [1]) objects. You can work in a way that mirrors your geometric intuition instead of getting lost in the mire of representational details.
[1] One of the real sticking points of trying to embed everything in a vector space is that a lot of the representations overlap, and you have to operate on them specially. Planes are represented as normal vectors, and so are regular vectors.