Let's start from the evolution side.
These are the things which drive evolution:
1. mutation
2. natural selection
3. the fact that a population cannot be infinitely large
4. the fact that not all members of a population breed equally well, or have the same number of offspring
5. mating between individuals is not completely random
6. populations can face disturbances that range from changes in environment, to individuals migrating in and out.
The Hardy-Weinberg equilibrium states, simply, that in the absence of the "disturbances" listed above, the frequency of each genetic allele in the environment ought to remain exactly the same from one generation to the next. The mathematical/probabilistic statement is based on the fact that Hardy and Weinberg used the formal laws of probability to demonstrate that in the absence of the factors listed, frequencies -will not- change. In other words, evolution will not occur, in any way.
The observable facts, however, are that (a) one or more of these factors *always* occurs, and (b) as a result, we can see changes in the frequency of alleles. That, too, was demonstrated mathematically by Hardy and Weinberg.
The simplest formal definition of evolution is simply "the change in the frequency of alleles in a population over time." That is, in fact, what evolution really boils down to.
There's more detail at
http://anthro.palomar.edu/synthetic/synth_2.htm .
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