The significance is explained in all the leading textbooks on genetics and evolution. I've chosen the explanation given by Carl Zimmer and Douglas Emlen because I know that Carl has spent a good deal of time getting it right in his new book Evolution: Making Sense of Life.
Imagine that you have a population with two alleles, A and a, at a single locus. The frequency of the first allele is f(A) to which we assign the value p. The frequency of the second allele is f(a)=q. In a randomly mating sexual population the probability of an A sperm being produced is p and the probability of an a sperm being produced is q. Similarly, the probability of an A egg cell is p and the probability of an a egg cell is q. These probabilities, p and q, do not have to be equal.
We can calculate the probabilities of all possible combinations or sperm and eggs in the population from a the following diagram (Punnett square). This one is from Wikipedia.
Since the total probability has to equal one, we have ....
This is the Hardy-Weinberg equation or the Hardy-Weinberg Equilbrium. What does it mean? Let's quote Zimmer and Emlen (page 156).
p² + 2pq + q² = 1
Hardy and Weinberg demonstrated that in the absence of outside forces (which we describe later), the allele frequencies of the population will not change from one generation to the next. As we'll see below, this theorem is a powerful tool for population geneticists looking for evidence of evolution in populations. But it's important to bear in mind that it rests upon some assumptions.There you have it. The Hardy-Weinberg describes the situation where evolution DOES NOT HAPPEN and thus serves as the null hypothesis for testing whether evolution is happening. Every undergraduate knows this.
One assumption of the model is that a population is infinitely large. If a population is finite, allele frequencies can drift randomly from generation to generation simply due to chance variation, in which alleles happen to be passed on to the next generation. (We will explore genetic draft in detail later in the chapter.) While no real population is infinite, of course, very large ones behave quite similarly to the model. That's because variation due to chance is inconsequential, and the allele frequencies will not change very much from generation to generation.
The Hardy-Weinberg theorem also requires all of the genotypes of the locusts are equally likely to survive and reproduce. If individuals with certain genotypes produced twice as many offspring as individuals with other genotypes, for example, then the alleles that these certain individuals carry will comprise a greater proportion of the total in the offspring generation than would be expected given the Hardy-Weinberg theorem. In other words, selection for or against particular genotypes may cause the relative frequencies of alleles to change and results in evolution.
Yet another assumption of the Hardy-Weinberg theorem is that no alleles enter or leave a population through migration. This assumption can be violated in a population if some individuals disperse out of it or if new individuals arrive. The model also assumes that there is no mutation in the population, because it would lead to new alleles
In each of these four cases, the offspring genotype frequencies will differ from the equilibrium predictions of the Hardy-Weinberg theorem. That is, because they alter allele frequencies from one generation to the next, selection, migration, and mutation are all possible mechanisms of evolution.
The Hardy-Weinberg theorem is useful because it provides mathematical proof that evolution will not occur in the absence of selection, drift, migration, or mutation. By explicitly delineating the conditions under which allele frequencies do not change, the theorem serves as a useful null model for studying ways of allele frequencies do change. The Hardy-Weinberg theorem helps us understand explicitly how and why populations evolve. By studying how populations deviate from the Hardy-Weinberg equilibrium, we can learn about the mechanisms of evolution.
Let's see if the Intelligent Design Creationists know this. I'm quoting "niwrad" from a post on one of the leading ID websites, Uncommon Descent: The equations of evolution.
For the Darwinists “evolution” by natural selection is what created all the species. Since they are used to say that evolution is well scientifically established as gravity, and given that Newton’s mechanics and Einstein’s relativity theory, which deal with gravitation, are plenty of mathematical equations whose calculations pretty well match with the data, one could wonder how many equations there are in evolutionary theory, and how well they compute the biological data related to the Darwinian creation.As you can see, the Intelligent Design Creationists interpret the "Hardy-Weinberg law" very differently, I wonder who is right?
The Hardy-Weinberg law mathematically describes how a population is in equilibrium both for the frequency of alleles and for the frequency of genotypes. Indeed because this law is a fundamental principle of genetic equilibrium, it doesn’t support Darwinism, which means exactly the contrary, the breaking of equilibrium toward the increase of organization and creation of entirely new organisms. To claim that the Hardy-Weinberg law explains evolution is as to say that in mechanics a principle of statics (immobility) explains dynamics (movement and the forces causing it).
So the initial question, how well math support Darwinian evolution, has the short answer: it doesn’t support evolution at all. Despite of the pretension of evolution to be a scientific theory with the mathematical certitude of the hard sciences, properly the equations of evolution do not exist.
Let's check with Joe Felsenstein. He's an expert on population genetics so he should know. Read his decision at: Evolution disproven — by Hardy and Weinberg?.