The puzzle is, that the universe has a macroscopic baryon number--about one baryon per 5 cubic meters, averaged over the volume of the universe. Put another way, about one particle in 10^10 in the universe is a baryon, and essentially none are anti-baryons. (Presumably there is also a net abundance of leptons--we see all these electrons but no anit-electrons. But neutrinos are leptons and are very hard to see, so we don't have any hard evidence that there is an imbalance one way or the other for leptons.)
The puzzle is all the more puzzling, because no laboratory experiment has ever observed baryon number to be violated, that is, in laboratory we have always observed that the creation or destruction of a baryon is associated with the creation or destruction of an anti-baryon. In fact, large experiments such as Super-Kamiokande have looked in vain for the decay of the proton, setting limits in the 10^33 year range for its lifetime. Yet our best theories of the origin (or at least early days) of the universe, such as inflation, suggest that it should have begun without any net abundance of baryons (zero baryon number).
Sakharov argued in the 1960's that any explanation of this puzzle had to contain three elements:
There are many proposals on the "intellectual market" for explaining how the baryons in the universe arose, but almost all of them require extensive speculative physics, generally involving energy scales which are far beyond the reach of our accelerators and which we are unlikely to have access to any time soon. Therefore, most of the proposals are essentially untestable, at least "in laboratory" (there may be ways of testing them by finding other cosmological predictions and checking those out with cosmology, but even this is not always the case). Electroweak baryogenesis is an elegant proposal, pioneered by Michael Shaposhnikov and collaborators in the late 1980's, which tries to explain the baryon number abundance using only physics which we know or will be able to investigate in particle physics experiments to be conducted in the foreseeable future.
The general idea is the following. It turns out that, even though no one has observed baryon number violation, the "minimal" theory of modern particle physics, the Minimal Standard Model, does predict that it should occur. (The Minimal Standard Model basically contains all the physics which HAS to be there to explain all observed particle physics, and nothing extra. Whatever the true theory of particle physics is, the Minimal Standard Model should be a subset of it.) It occurs via something called a "sphaleron transition" (a name invented by Nick Manton in 1984, who pioneered the idea), which turns out to be a phenomenon which is extremely inefficient under normal conditions (so we should not have observed baryon number violation), but becomes efficient at very high temperatures (meaning 100 GeV, or about 10^15 Kelvin if you prefer). I will explain how this works here. Furthermore, the Minimal Standard Model contains CP violation -- which it had better, because CP violation has been observed in several laboratory experiments. Finally, there is the possibility of a departure from equilibrium at the Electroweak Phase Transition, which I explain in a bit more detail here.
I worked for a few years on this general problem; I was particularly involved with trying to understand the "sphaleron processes" by which baryon number is violated. However, work in this field has generally died down and I am not actively involved in it and have not been since late 2000. The reasons are, first, that it is appearing less likely that WEBG is the right explanation to the origin of the baryon number abundance, and second, that the open questions are more and more becoming experimental high energy physics questions which can only be answered by experiments which will not run for a few years yet. The remaining theoretical questions to pursue are now mostly either highly model dependent (that is, they depend sensitively on what physics beyond the Minimal Standard Model actually exists), or extremely "grubby."
There are two big problems.
The first is that the CP violation of the Standard Model is far, far too weak to explain baryogenesis. There must be extra physics which introduces new CP violation. There are some strong limits on such new CP violation, which generally require it to occur via interactions which will be very hard to measure in future particle physics experiments. In any case, the CP violation must involve new physics we don't know about.
The second is that, in the Minimal Standard Model, now that we have pretty firm lower limits on the mass of the Higgs boson (which is still unobserved), we know that there is no electroweak phase transition. That does not mean that there is no electroweak phase transition in nature, because we don't know if the Minimal Standard Model is the complete description of nature. But for there to be an electroweak phase transition, there must be new undiscovered particles besides the Higgs boson. Furthermore, they must be relatively light and have quite large couplings to the Higgs field. (How light depends on how large the coupling to the Higgs field is. The heavier the extra particles, the bigger that coupling must be.) For instance, if the Minimal Supersymmetric extension to the Standard Model (MSSM) is correct, then the scalar partner to the top quark, the "stop," must have a mass lower than about 140 or 150 GeV. This is not experimentally excluded, but again it is an experimental question which we will not know the answer to for several years.