What's the Clifford algebra? I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really is? (Keep in mind that I don't know abstract algebra, nothing except some group theory.) 
Does it make sense to write the sum of a scalar and a bivector in the Clifford product? Both are very different things.
 A: The Clifford (or "geometric") algebra is an algebra "generated" by a vector space with a bilinear form with some special properties. There are several pockets of physicists and mathematicians experimenting and refining how to use these algebras to do some geometry.
If you like, you can think of the real 3-D space as your vector space $V$ with its regular inner product, and now imagine it sitting inside an algebra $Cl(V)$. (I am omitting a lot of details in the interest of simplicity.) Since $V$ is three dimensional, so you expect $Cl(V)$ to have at least that many dimensions, but it turns out it will be $2^3$ dimensional (and $2^n$ dimensional for an $n$ dimensional $V$.)
Since $V$ is inside of an algebra $Cl(V)$ there is a multiplication defined between every two elements of $Cl(V)$, and in particular between any two elements $x,y$ of $V$. In fact, things are set up so that $v^2=v\cdot v$, where the right hand side is the inner product in 3-space, and $v\in V$.
Clearly there are many more elements in $Cl(V)$ than just those in $V$. The question is: do they have a useful interpretation? The answer is "yes", because of the way things are set up. It turns out that subspaces of $V$ can be represented as products of elements of $V$ in $Cl(V)$. Because multiplication is defined everywhere, you can now "multiply things in $V$ with subspaces of $V$".
Again, because of how things are set up, you can use this multiplication to do a lot of useful things. For example, you can determine if an element of $V$ is in a certain subspace, or you can reflect a vector in a subspace, or do rotations about a vector in $V$, all using the product of this algebra (and a couple of derived products which I'm not mentioning.)
I'm going to throw in here that when you stop using the usual inner product on $\Bbb{R}^3$, and you switch to the zero bilinear form (that just says "$x\cdot y=0$ for all $x,y\in V$") you get exactly the exterior algebra of $V$ another famous and useful algebra-geometry topic. 
That is a very skimpy introduction. I can recommend a lot of resources on the topic, though. If you are very physics oriented, then I think you will enjoy Macdonald's online pdfs on the topic. They are very easy to read. There is also a text (which I have browsed but not really read yet) by Doran and Lasenby called "Geometric algebra for physicists". Another good link, and even an MIT course,another good one: Lundholm-Svensson, good general info.
If you ever want to read it from an algebraic perspective, Clifford algebras appear in Jacobson's Basic Algebra volumes (one or both), and they also appear in Artin's Geometric Algebra. (However, be aware that Artin's title is meant to be taken literally, and it is not really referring to what Doran, Lasenby et al are referring to.)
You will find no shortage of information on "geometric algebra" online, since it is not really a popular visible topic, and the promoters must keep the information freely flowing. I say it's not popular "not very visible" in the pure math community, but it sure is an interesting and fun topic. I imagine as physicists and computer scientists continue to find applications for it, the pure math community will eventually get interested too.

By the way, I saw you noted "$ab=a\cdot b+a\wedge b$" and commented on the exterior product. The wedge there is one of the products that I skipped mentioning for now. It shares many properties with the exterior-algebra-exterior-product. 
The identity $vw=v\wedge w+v\cdot w$ works for $v,w\in V$ "because of how things are set up." Surprisingly, the identity also works if you put a few things other than vectors into it, too. 

I also thought I would give what I thought were the most elementary cases you might start looking at. Certainly 3-D space with the usual (+++) inner product is where to start. Later, you might get interested in the case where the signature of the inner product is (+---) or (-+++).
Before you do the 4-d space though, you might want to puzzle out how $\Bbb{C}$ and $\Bbb{H}$ are Clifford algebras for a 1 dimensional real vector space with signature (-), and for a two dimensional space with signature (--) respectively.
A: When I first read about 'geometric algebra' and then went down the rabbit hole into exterior algebra, Clifford Algebra and so on, I was struck by mixture of numbers and geometries. I was also excited to see that this was a different way of thinking and offers new insight to how physics and engineering can be taught.
Through most of STEM education, we are pushed up through calculus (all levels) and pick up a few statistics and linear algebra courses. All the while, there is the use of vectors to represent mechanics and fields. What geometric algebra (and Clifford Algebra) does is to extend these concepts and show the inner workings of a larger mathematical basis where vectors take on new meaning. For example, in engineering, we use the cross-product to multiply two vectors and obtain things like a torque vector from a force vector and radius vector. This is limited to 3D. When you use geometric algebra, you find that the ‘wedge product’ is applicable and it is not limited to 3D.
To understand how different geometries can be used in the same equation, you have to consider generalizing the vector. A vector in 3D is the addition of three independent components. Each component has a scalar number, $x$, associated with a unit vector, $v$. Note that the unit vectors keep the numbers separate. When written out, the resultant vector might look like
$$
R = x_1 [v_1] + x_2 [v_2] + x_3[v_3]
$$
In this equation, you would not add $x_1$ to $x_2$ as they are not associated with the same unit vector. Also, the unit vectors are each geometrically distinct. This follows vector algebra.
Another interpretation is that there are different types of addition. Numerical addition is when numbers are added together. Geometric addition is when geometries are added together. To be clearer in the introduction, the geometric addition should be a different symbol, such as circle-plus, $\oplus$, to visually distinguish it from numerical addition, $+$. If this was done, the equation above would be (circle-plus is used)
$$
R = x_1 [v_1] \oplus x_2 [v_2] \oplus x_3[v_3]
$$
This could be extended to higher dimensional vectors.
So if you are comfortable with adding different vectors together, then you can apply this idea to add different geometries together, such as points, vectors, bi-vectors, etc. There is a whole set of geometries that can be constructed by multiplying vectors (not as dot and cross products) with extended multiplication provided by the wedge product. With this start, you can begin to see the impact of numbers and geometric elements and how they can play a role in the same equations.
A: Let's consider a Euclidean 3d space.
The clifford algebra on this space consists of multivectors, which have 8 linearly independent components.  These components can be broken down as follows:


*

*1 scalar component

*3 vector components

*3 bivector components, which correspond to the 3 linearly independent planes in a 3d space

*1 trivector or pseudoscalar component, which corresponds to the single, oriented unit volume in 3d space


(Note the 1,3,3,1 pattern.  An N-dimensional space will have $2^n$ components for its multivectors, broken down in this way as according to Pascal's triangle.)
Because all of these components belong to a single multivector object, it does make sense to say you can add scalars and bivectors.  You just add corresponding components as you would with traditional vector algebra.
The clifford algebra of 3d space is the algebra of these multivectors--so you have addition and subtraction as usual.  Clifford algebra also makes possible various products of multivectors, which lend themselves to simple geometric interpretations.
Beyond that, if you have a more specific question, I'd be happy to try to answer it.  As far as saying what clifford algebra is (from the perspective of a lowly physicist), this is all I can say.  Why it's useful or how one uses it on a practical level is another question entirely.
A: From a 'what makes it tick' viewpoint, here is a function that generates Clifford Algebras with any signature, in pseudocode:
for nbr_generators = 0 to Whatever \n
max = 2^n - 1
for signature = 0 to max   // all possible signatures +++ to ---...
   for i = 0 to max        // multivectors in LEXICOGRAPHIC order !
      for j = 0 to max 

         k = ( i XOR j )
         sign_ac = anticommuting_sign( i, j )
         sign_sig = signature_sign( i, j, sig )
         net_sign = sign_ac * sign_sig 
         Result[ k ] = A[i] * B[j] * net_sign  // the product
         Cayley_table[i,j] = net_sign 

signature_sign( i, j, sig )  // bit is ON if generator squared = -1 
  squares = ( i AND j )
  negative_squares = ( squares AND sig )
  if ( bitcount( negative_squares ) ) is odd
     return -1
  return 1

anticommuting_sign( i, j ) 
  convert i and j to strings - each ON bit to a letter
  concatenate string_j to string_i
  do a bubble sort counting exchanges of distinct letters
  if nbr_exchanges is odd
     return -1
  return 1

