What exactly is a kernel?

There is no real connection between the null space/kernel of a matrix and the other two meanings you cite. This use of kernel is a special case of the more general notion of the kernel of a homomorphism between two algebraic structures.

Mikio L Braun has already pointed out the relationship between the other two uses you asked about. You might also try browsing the computing and mathematics sections of Wikipedia's kernel disambiguation page.

I'm not sure about the null space, but I guess since algebra is quite, well, abstract, people were just taking whatever made sense to them.

Now concerning the connection between Mercer kernels and convolution kernels is through integer operators. Integer operators are a class of mapping between function spaces studied in the context of differential and integral equations. An integral operator maps a function f(x) to a new function g(x) = "integral of k(x, y) f(y) dy". The function k(x,y) is called the kernel. It can be thought of as an generalization of a matrix.

Under mild conditions (which I don't recall right now) integral operators have nice properties with respect to their eigenvectors and eigenfunctions, namely, that there are only finitely many eigenvalues and the only near point is zero. In particular, if the kernel fulfills Mercer's condition (i.e. is symmetric, and positive definite), then there exists a series of non-zero eigenvalues and orthogonal eigenfunctions which lead to the well-known expansion as a series.

Now, the convolution of f with g is given by the integral of g(x-y)f(y) over y. In other words, the convolution has the same form as an integral operator with kernel k(x,y) = g(x-y).

There's also kernels as factors in boolean expressions. I'll paste below some old text I wrote summarizing the definitions - it gets to kernels in the third paragraph. Sorry about the length, but might as well leave the terminology clear.

Kernel extraction is a method for factoring boolean expressions introduced by Brayton and McMullen in [Brayton82], later used in the Mis logic synthesizer and more extensively documented by Brayton et al. in [Brayton87].

A variable represents a single coordinate of the Boolean space. A literal is a variable or its negation (e.g. a or a'). A cube is a set of literals, defining a subset of the Boolean space (e.g. {a,b',c}). An expression is a set of cubes (e.g. {{a},{b,c'}}); it can be used to represent (not uniquely) a logic function, in a sum-of-products form (e.g. a + bc').

Given a definition of division where, for example, (ab + bc) / b = a + c (no remainder) and (ab + c) / a = b (with remainder), an expression is cube-free if no cube divides the expression evenly (ab+c is cube free; ab+bc, abc are not). The primary divisors of an expression are the set of expressions that result of dividing the expression by a cube. A kernel is a primary divisor that is cube free; its co-kernel is the cube used to divide the expression.

For example, x = (a+b+c)(d+e)f+g has a kernel a+b+c, corresponding to co-kernels df and ef (i.e. this kernel is obtained by x/df or x/de); d+e is a kernel corresponding to co-kernels af, bf, or cf; (d+e)f is a primary divisor (x/a, x/b, or x/c), but it is not a kernel because it is not cube-free.

Finally, the concept of level of a kernel is recursively derived from a kernel containing another kernel of the same expression. A level-0 kernel does not contain any other. For example, if a(b+c)+d is a kernel of an expression, it will be a level-1 kernel, because it contains the level-0 kernel b+c, which cannot contain any other.

[Brayton82] R. K. Brayton and Curt McMullen. The Decomposition and Factorization of Boolean Expressions. In Proceedings of the Int. Symp. Circuits and Systems, pages 49–54. IEEE, 1982.

[Brayton87] R. K. Brayton, R. Rudell, A. Sangiovanni-Vincentelli, and A. R. Wang. MIS: A Multiple-Level Logic Optimization System. IEEE Transactions on Computer-Aided Design of Integrated Circuits, CAD-6(6):1062–1081, November 1987.

I can answer the null space part

in linear, suppose you have a matrix A m-by-n.

the null space or kernel space of A ( null(A) ) is the set of all vectors x which satisfy this equation

A.x = 0