Since I am a HETP, let me try to help out here a little, and see if I can't clear up the confusion caused by this incredibly poorly written BBC article....
Modern cosmological/particle physics tells the following story: the universe started from a singularity, with physics we don't understand. For some reason, it began to expand (the big bang), and the energy that was liberated was able to split into particle/anti-particle pairs. These pairs would annihilate again into radiation, and the cycle would continue, over and over. Now, in the absence of any interesting physics (which I'll get to later), as the universe cooled down, there would come a time when the radiation was no longer energetic enough to produce new pairs, so the ones that existed at that point would (more or less) all annihilate. Since we started out with none, and they are only created in pairs, we would end up with a net concentration of zero. Hence the problem: we don't have zero particles, we have lots of them. Of course, they wouldn't ALL be able to find each other (hence, the "more or less" above). If we ended up with a few more particles over here due to random motion after pair creation, there would be a few more anti-particles over there. As the universe expanded, these "islands" would move apart. Two problems: first, no one has come up with a reasonable mechanism to produce this matter/anti-matter separation (why would electrons and protons all end up in the same islands, and not end up in overlapping, but distinct islands?), and there are predictions as to how much radiation would be coming from the "domain wall interface" that separates islands. And we don't see that radiation, despite the fact that we should be able to detect islands of approximately the size of the visible universe.
So, along comes Andrei Sakharov, and he points out that there could be a mechanism that creates more matter than anti-matter, and it requires only three ingredients: 1) baryon violating processes (processes that create, net, more baryons in the final state than in the initial state of a reaction), 2) CP violation (a difference in the properties of matter and anti-matter, and 3) regions of the universe that are not in thermal equilibrium. It turns out that it is relatively easy to produce models with all of these properties, so it is relatively easy to produce models that generate more matter than anti-matter in the universe (In particular, the Standard Model of particle physics has all of these properties). The hard part is building a model that predicts a scenario that matches the universe we observe. Notice that CP violation is a major component of this story, so understanding it in detail helps us build models that more closely approximate the universe.
So what is CP violation? The details are technical, and require lots of equations, but the simplest description is that the laws of physics would not be quite the same if time ran in the opposite direction, from future to past. "Well, Duh!!" you might say, since our everyday lives suggest that this is the case (after all, if you took a home movie and played it backwards, everyone would be able to tell that it was running backwards). But physics on the subatomic scale, was believed until the mid 60s to run both forward and backward identically. (The perceived asymetry on the macroscopic scale is a consequence of large N statistics, not microphysics. This statistical mechanics asymmetry of time exists even in a fully CP invariant theory).
Now, to what these experiments measured: there are many possible sources of CP violation, and separating them out requires exquisitely sensitive experiments...just proving that CP is violated is extremely tough! The CERN and FNAL experiments were, quite embarrasingly, far out of agreement a few years ago, owing to the difficulty of making the measurements. Although I haven't read their current papers, I suspect that after analyzing more data, they are now closer to agreement with each other. And that is a major leap forward in our understanding of the CP violation parameters and the origin of the matter/anti-matter asymmetry of the universe.