I'll try to make this easy, although it's hard, and I don't understand it all, so that's working against us all here too.
Basically, the Standard Model of Particle Physics (a big huge complex math equation used for predicting anything... distances, velocity, mass, etc)
doesn't work that well really. That is... unless we include a "symbol" for this Higgs Boson that no one had ever found. We knew it had to be there, because the math equation wouldn't work without adding that symbol (and all it's inherent properties=smarty part). This is what lead to the Theory and subsequent search for it. Just the fact that the math being done to back up observations being made in any given field, was infinitely more complex and unsymmetrical without adding the values of the Higgs.
Now they have verified it's existence and thus the Standard Model. We know that our math matches up as concisely as we're able to conceive and test to this point. That's a pretty big deal in of itself, but it gets better, because now this can become a hugely funded area of research.
So that's why they have been searching essentially. If they didn't, everyone everywhere was going to have to redo their math, and make it all a whole fuck-ton harder and rigid.
But the Higgs Boson itself "proves" something cooler too. The existence of the aptly named Higgs Field. The Higgs field permeates the entire universe, like an electric field or magnetic field, etc. And like an electric or magnetic field certain particles interact with it more strongly or weakly dependent upon it's relative aspects. So if a particle has a very strong interaction with the Higgs Field, it will slow down dramatically, become heavier, gain mass. If it has a weak interaction (or no interaction) with the Higgs Field, it will pass right through everything, gaining neither mass nor losing velocity. An example of a a weakly interacting particle is the Neutrino
So it can almost be thought of like a charge. We know particles can have an electric charge, positive or negative. They can also be charged in a way where they interact with the Higgs Field, or don't. Interestingly, it doesn't seem like you can have a "negative" Higgs charge though.
The practical implications of this will be huge. But just the fact that we can actually move past verification and start teaching new minds about these principles will be the biggest difference. The research that is going to go into this field now will be unbelievable. And think about it, we're talking about breaking down and gaining an understanding of the cosmic field ultimately responsible for Gravity and
Mass. Gravity manipulation? Hover board? 2015?
/not that I'm making any predictions or anything but..