Currently, things weigh more than they should. The mass of a particle is a function of the kinetic energy of the particle and it's component parts, if any. If we run the numbers, we get good masses for some particles, not good masses for others. A proposed solution to this problem is the Higgs field, a nonzero field that permeates space. Anything coupling with this field gains additional mass through interaction with the field.
Picture a person at a party. Normally, they are free to move through the party fairly easily. Now make that person famous. Admirers flock around, and the celebrity has trouble moving. Nonfamous people are particles that do not couple with the HIggs field. Celebrities are particles that do couple with the field, surrounded by a paparazzi of virtual Higgs particles.
Nice theory. It fills a gap in the standard model and now the math all works. So now we have to find the particle. You need the mass of a particle to find it in an accelerator. Roughly (very roughly), you need to create collisions where the sum energy of the little explosion is about that of the particle in question, then watch a statistically large number of those to see if something matching your particle appears. If it does, it's off to Stockholm for dinner with the king. If not, it's back to the drawing boards.
The problem is, the theory doesn't predict the mass of the particle. It doesn't even say if it is one particle, a family of similar particles or a family of different particles. So there's a wide spread of masses to examine. And all the masses are really high, far higher than any other existing accelerator could reach. So we have the new CERN experiment, slowly scanning the possible masses, looking for the particle.
If we don't find that particle, then we're back to square one, why are some particles heavier than predicted? For decades, we've assumed it was some sort of variant of the Higgs boson. But if that's not the case, it's back to the blackboard for more theories.
In general, this is a problem for particle physics. Finding or not finding the particle will affect chemistry, biology and general astronomy not at all. It might or might not have an affect on cosmology, but that's hard to say without a particle to talk about. More interesting for cosmology is that while searching for the Higgs, the experiment might come across more esoteric things, such as evidence for supersymmetry. Evidence for supersymmetry would automatically generate the prime number one candidate for dark matter. And nailing down the properties of dark matter would give us another probe of the Big Bang.
More information than you wanted probably, but I hope it helps.