So to make this familiar, you're probably used to traditional coordinate vectors in geometry. For example a 3D vector [x, y, z]. This seems sane enough, but is actually somewhat ambiguous. Which cardinal direction map to x, y, z respectively (e.g. is z for up/down or forwards/backwards)? What direction do the axes point, e.g. is +z away or towards us? To make it unambiguous, we can define three oriented basis vectors. That is, each basis vector defines what the unit is in that direction, and which way is "positive". So I could write [3, 2, 9] as 3*xunit + 2*yunit + 9*zunit.

Now in geometric algebra we also have oriented basis vectors. So they determine the unit, and the direction of the embedding space (note that this not necessarily map directly to points in your world space, which becomes especially clear when you start doing projective geometry, e.g. in 3D PGA you represent lines in your world as planes in the embedding space). Rather than xunit, yunit, etc, they are simply called e1, e2, ..., en.

The key difference is that geometric algebra has the exterior product, notated ^. For example, e1 ^ e2 is the exterior product of two oriented basis vectors. You can interpret this as being an oriented basis for the space spanned by e1, e2. And similarly for e1 ^ e2 ^ e3 being an oriented basis of the volume spanned by e1, e2, e3, etc. These are called basis blades.

The most general closed numerical representation of an object in geometric algebra is the multivector. Just like a traditional vector is just a bag of K coefficients for K basis vectors, a multivector is a bag of 2^K coefficients for the 2^K basis blades, one for each possible combination of the exterior product of the K basis vectors.