I started learning about Clifford algebra recently. Tensors are familiar in physics; the algebra of tensors is built up from vectors using the tensor product, and it includes the vectors as a special kind of tensor. Similarly a Clifford algebra is an algebra built up from the vectors, but using a different product called the Clifford (or geometric) product. A Clifford algebra contains the vectors as well as more general elements (comparable to higher rank tensors) called multivectors.
Given any vector space we can build a tensor algebra from it. But to build a Clifford algebra we need to be given an inner (that is 'dot') product along with the vector space. Pure mathematicians (e.g. Chevalley, E. Cartan) have studied Clifford algebras in the abstract, with arbitrary vector spaces, fields and inner products. The corner of Clifford algebra known as geometric algebra deals mainly with 'ordinary space,' i.e. R^3 with the ordinary dot product, and 'Minkowski space,' i.e. R^4 with the inner product familiar from special relativity. The Clifford algebras in each case can be used much like the tensor algebras; you can use multivectors, like tensors, to represent things which vectors don't suit, like the electromagnetic field.
There are several books that I've found really helpful (I'm a beginner).
Pertti Lounesto's
Clifford Algebras and Spinors starts right from the beginning, introducing vector spaces before building up the Clifford algebra in 2-D. The first half of the book introduces Clifford algebra and some physical applications, the second half covers more advanced mathematical topics (I haven't looked at the second half). It is fairly concise, and has lots of exercises with answers, which include lots of chances to get used to actually calculating things and a few proofs too.
Geometric Algebra for Physicists by Doran and Lasenby, mentioned above, is also very useful. It contains a free standing introduction to Clifford algebras, as well as 'geometric calculus,' which is (multivariable) calculus with Clifford algebra. It covers a wide variety of physical applications including point and rigid body mechanics, electromagnetism, and spinors in quantum mechanics. The applications chapters aren't stand-alone introductions, and as mentioned above they probably won't make sense unless you know the physics before.
For a summary of the maths
A Survey of Geometric Algebra and Geometric Calculus by Alan Macdonald (available here
http://faculty.luther.edu/~macdonal/ ) is very clear.
The (partly unfinished) notes
Clifford algebra, geometric algebra and applications of Lundholm and Svensson, found here
http://arxiv.org/abs/0907.5356, are much more mathematical and looked a bit intimidating to me, but they define the various products of Clifford algebra in a very simple way which allows you to prove identities essentially by verifying obvious Venn-diagram type set theory relations. They also treat duality, and how it relates the various products, very clearly (these duality relations speed up derivations a lot).
Hestenes'
Spacetime Calculus, here
http://geocalc.clas.asu.edu/html/STC.html, is a concise introduction to Clifford algebra in relativity, I found the section on electromagnetism very helpful. I haven't looked much at Hestenes'
New Foundations for Classical Mechanics, and Hestenes and Sobczyk's
Clifford Algebra to Geometric Calculus.
As an aside, some books on geometric algebra are written from the point of view of trying to establish geometric algebra as the 'universal mathematical language' or 'a grand unifying nexus for the whole of mathematics' (quotes from Hestenes website
http://geocalc.clas.asu.edu/ ). This involves reformulating everything in terms of geometric algebra and criticising other ways of doing things. As a result these books can sound a bit 'cult-like', which can be off-putting. It is not true that tensors and differential forms lack geometric interpretation; looking in the 1951 edition of 'Tensor Analysis for Physicists' by Schouten (one of the founders of tensor analysis), you find a glossy page with photos of wire and foam models of geometric representations of various kinds of tensor! Emphasising geometric interpretation is helpful, but not as revolutionary as you'd guess from reading some geometric algebra books. Also, the conviction that everything must be done using geometric algebra leads to squeezing subjects like differentiable manifolds into a formalism that doesn't seem to suit them (to me).
Geometric algebra does suit some things extremely well though, for example:
(i) Describing spinors, and understanding the Pauli and Dirac matrices as representations of basis vectors in space and spacetime respectively. This speeds up calculations and makes some Pauli and Dirac matrix identities obvious. For example the Pauli matrix identity (\vec{x}\cdot\vec{\sigma})(\vec{y}\cdot\vec{\sigma}) = (\vec{x}\cdot\vec{y})I+i\vec\sigma\cdot(\vec{x} \times \vec{y}) expresses the simple GA relation \vec{x}\vec{y} = \vec{x}\cdot\vec{y}+\vec{x}\wedge\vec{y} via a matrix representation using the 'vector of matrices' \vec\sigma.
(ii) Doing electromagnetism. Maxwell's equations reduce to a single GA equation \nabla F = J and, more importantly, in GA \nabla can be inverted (i.e. has a Green's function), so you can solve for the 'field tensor' F in terms of the four-current J directly without introducing potentials. As another example, Coulomb's law in electrostatics follows simply from \vec\nabla\cdot\vec{E}=\rho and \vec\nabla\times\vec{E}=0, since together these can be written \vec\nabla\vec{E}=\rho, and we can invert \vec\nabla by integrating with its Green's function.
In short, while some people get overzealous, Clifford algebra is frequently very useful.
