Yes, it almost certainly will. It will tell us more about the behaviour and properties of elementary particles, and hopefully something unexpected and interesting. That way we will understand the fundamental laws of nature better.
These laws of nature will also be very important for understanding the early universe and therefore also explaining the properties of the universe today. This is because we know that the universe was very dense and hot when it was young, and processes that are studied in the LHC, would have been very common in the early universe. One particular question that I hope the LHC will help us answer is why there is so much matter in the universe but practically no antimatter at all, but it may also help us answer many other important questions.
At a minimum we expect to answer the question of what is the mechanism by which electroweak symmetry is broken. Electroweak symmetry means that photons of light are fundamentally the same as the W and Z bosons that carry the weak nuclear force. The W and Z have a mass though – we call this a broken symmetry. It is hard to explain this in the Standard Model of physics as we know it now, because the masses cause some nonsensical predictions – for example the probability for two W bosons to scatter off each other becomes greater than one (this is like saying you could through a single dice and score two sixes!). Since this doesn’t make sense, there must be some new phenomenon that causes the mass in a natural way.
Depending on what the answer to the electroweak symmetry breaking turns out to be, there may also be other new things like dark matter candidates, or something completely unexpected.