We can use matrices to represent rotations without going in…

We can use matrices to represent rotations without going into Clifford algebras, so that's kind of a red herring. A clifford algebra has to do with quantum mechanics, and it has to do with the fact that the probability of an observation is proportional to the square of the wave function. It turns out that spinors can become relevant in this context because they can act as square roots of vectors. As with ordinary square roots, there is a negative and a positive root that correspond to the same rotation. This is what the "double" in double cover refers to. Clifford algebras are a mathematical idea that can be used to generate spinor states. Spinors can also be represented by matrices but they act on a complex vector space, whereas rotation matrices act on a real vector space.