How did they discover the Higgs boson? But first....
Matter is made up of molecules - which are made up of atoms - which are made up of sub-atomic particles - some of which are considered elementary particles by the Standard Model of particle physics. The Standard Model is a model in the same way a two-dimensional map is considered a model of the real world. The model isn't correct. In fact it is necessarily wrong (as quoted by a famous statistician). But it is useful. For decades physicists hoped that the Standard Model would be useful in explaining what happens in the tiniest spaces and biggest galaxies around us. In addition to explaining the world, the Model makes certain predictions. One of them is the existence of the Higgs boson. By discovering it, physicists have greater confidence that the Standard Model may actually be somewhat less incorrect.
Back to the discovery. Two things to keep in mind: to find the Higgs boson, physicists need to smash together protons traveling at very high velocities towards each other and look through the debris of the collision; and the Higgs boson spoils very quickly. So quickly in fact that physicists can never see it or record it directly.
The first fundamental difficulty is that the protons have to smash into each other with sufficiently high energy to possibly spew out their Higgs bosons. How fast do they have to go? Almost the speed of light. To make that happen, the human race built the most expensive scientific instrument ever. It was supposed to be built in Texas (Waxahachie) but the US Congress axed that project in 1993 setting human knowledge back by a couple of decades and moving the epicenter of particle physics to Europe. It was eventually built on the France-Switzerland border and cost about $5 Billion and is called the LHC (the Large Hadron Collider). It uses an underground tunnel and magnets to focus two beams of particles traveling in opposite directions on to each other. Particles, like billiard balls, prefer to travel in straight lines. Magnets placed along the path can curve the trajectory of the particles and force them to travel in a loop. The faster you want to sling the particles, the better the magnets need to be. Colder magnets are more powerful than hotter ones. The magnets at LHC are cooled by one of the coldest substances known to mankind - liquid helium - to almost the coolest temperature possible theoretically. The LHC can smash two opposing protons at 7 Tev (7 million million electron volts). That is approximately the energy of a flying mosquito. Now imagine all the molecules and atoms and protons in a mosquito. Take just two of those protons and give them all that energy. 7 Tev is a lot of energy - the most we have ever been able to muster to smash two protons into each other.
The second fundamental difficulty is that you can't detect the Higgs boson directly because it is too short lived. Just as it is hard to photograph a child that won't sit still, we don't yet have the technology to detect and record a Higgs boson directly. What is a physicist to do? Imagine you don't know if you have any eggs in your fridge. To find out you decide to split its contents equally into two ziplock bags. Then you load the ziplock bags into guns and shoot them towards each other. As they collide mid-air you photograph the mayhem. And from analyzing the pictures of the resulting splashes you determine that there was an egg in the fridge. In much the same way physicists look for Higgs bosons. They know that if the Higgs boson was to exist, it would disintegrate into some specific set of other particles with a specific probability and some of those particles could be detected some of the time. There would also be a lot of other shit flying around. But if you looked through enough collisions, you should be able to tell if there were any Higgs bosons in your fridge. Scientists at the LHC sifted through collision data from 300 trillion proton-proton collisions before they had the statistical evidence to say they had discovered it. Also equally amazing is the work done by scientists at the American Fermilab Tevatron collider. The collider closed it's doors on September 2011, but by analyzing data from years of experiments from the past, scientists were able to say with great certainty (but not with as much as the LHC scientists) that the Higgs boson does exists and what it's physical description approximately is. They postulated this days before the final announcement from LHC.
So now we have the Higgs boson. Big questions remain. Who knows how many of them will be answered before it is time for your high-school physics class.
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