GA is a Clifford algebra that has some nice properties for working with geometry. The most notable fundamental difference is the presence/use of the "wedge" or outer product, which gives you the region created by sweeping one vector along another - i.e. (1,0,0) wedge (0,1,0) defines a 1x1 region in the XY plane, with an orientation i.e. the direction that it is facing is implicitly defined by the order in which you wedge the vectors, without need for a separate normal.
Of course in order to express that we're not dealing with regular vectors anymore... instead we go to multivectors, which (for R3) contain a scalar, 3 vectors akin to x, y, and z, 3 bivectors akin to xy, yz, and xz, and 1 pseudoscalar or "directed volume element" akin to xyz. To some extent the coefficients of the bivectors can be considered the 'projection' of the vector onto those planes. The pseudoscalar's weird though.
The geometric product of two multivectors is defined as the inner ("dot") product plus the outer ("wedge") product. Though it's hard if not impossible to visualise, the geometric product allows us to multiply vectors together (and divide them).
Those are the basics. Then it starts getting weird and I have to review the powerpoints they gave us.