Clifford Algebras

Clifford algebras arise in several mathematical contexts (e.g., spin geometry, abstract algebra, algebraic topology etc.). If you're just interested in the algebraic theory, then the prerequisites would probably be a solid background in abstract algebra. For example, I think linear algebra and ring theory are prerequisites but in practice, one should probably know more (e.g., for motivation and mathematical maturity). If you could elaborate further on your mathematical background, then I'm happy to provide more detailed suggestions.

I think this link provides a nice elementary introduction to Clifford algebras: http://www.av8n.com/physics/clifford-intro.htm. If that's too basic for you, then also have a look at: http://www.fuw.edu.pl/~amt/amt2.pdf. If you're familiar with algebraic topology, then the following paper is very interesting: http://www.ma.utexas.edu/users/dafr/Index/ABS.pdf.

When first starting out, I found some of Alan Macdonald's introductory material to "geometric algebra" very useful for developing intuition.

To make a gross overgeneralization, geometric algebras are basically the low dimensional Clifford algebras over $\Bbb R$ that are most relevant to 2-d and 3-d geometry, and even some 4-d relativistic geometry. I think it's beneficial to have that experience before seeing more general Clifford algebras over different fields, with higher dimensions, with different forms, etc.

I found that paper and several of the other papers he has online very helpful. There is a more technical description, along with some practical uses, in Jacobsons Basic Algebra II.

Slightly more physics-y versions of the same content are found here. I have not had the opportunity to read it, but Lundholm's material is also something I see frequently suggested.