This gives a very helpful geometrical description which finally let SVMs make sense to me. The weights are a vector normal to a family of planes and the optimization finds the two parallel planes that most separate two categories of data.

Solving the optimization is performed in terms of the inner product of data vectors. This inner product can be replaced by a function of the inner product (the kernel) in order to transform the data which may otherwise overlap into a space where a separating plane may be found.