Here's my best shot at explaining what they found. First, let's go back to middle school science and remember that everything is made of atoms like hydrogen, carbon, oxygen, nitrogen, silicon, neon, gold, etc. All of these atoms are made up of particles and the three particles you probably learned about are protons, neutrons, and electrons, of which electrons are the smallest. It turns out that there are even smaller particles (called subatomic particles) in atoms. These subatomic particles make up protons and neutrons and do some other things in the atoms as well. Examples of subatomic particles are quarks, muons, and bosons. The Higgs boson is one of these subatomic particles that was named after the scientist who theorized about it (Peter Higgs).
Basically, what Peter Higgs noticed is that light (which comes in units called photons) moves through space really, really, really fast (around 700 million miles per hour) and particles like electrons can also move through space really, really fast, but not quite as fast as photons can. Higgs figured that there must be something slowing down these electrons and other particles that isn't slowing down light. He came up with this idea that space (which we previously thought was empty) is filled with a "field" of these subatomic particles called bosons that create some type of friction-like force that slows down particles that have mass. The greater the mass of the particle is, the more these bosons slow down the particle. That would explain why a photon, which has effectively no mass, can travel so much faster than particles that have mass. He called the field the Higgs field and the particles that create the field Higgs bosons (because scientists love naming stuff after themselves, see: the Hardy-Weinberg principle, the Pythagorean theorem, Halley's comet...although making things even more complex, often things get named after the person who wasn't the first to discover them).
So what was actually discovered a few days ago? Well, in the Large Hadron Collider, scientists speed atoms up to a terrific rate and crash them into each other. When they do this, the atoms burst into all of their component pieces of subatomic particles. A few days ago, scientists collided some atoms and recorded evidence of a new type of never-before-seen subatomic particle that they think could be the Higgs boson, which until now was just theoretical. And that's what has everyone so excited. We're still not 100% sure that what was discovered was a Higgs boson, but it was definitely some type of new subatomic particle, and that's pretty exciting.
For a visual explanation, this video does a pretty good job.