This can be something of a confusing question, because the answer can be either one of two directly related phenomena, and both involve movements in the Earth's crust. An earthquake is the ground motion you experience when a part of the Earth's crust shifts along a surface, a sort of fracture in the crust, known as a fault. This shift produces vibrations in the rocks and soil beneath us. These vibrations are called seismic waves, and if they are large enough, they can be felt, just the same way that you can feel a bell vibrate when it is rung. So when you feel that kind of shaking of the ground, you could say "That was an earthquake!", and you'd be correct.
On the other hand, not all shifting along faults results in seismic waves strong enough for a person to feel. The smallest of shifts often aren't even recorded by sensitive instruments. Yet every shift is fundamentally the same, no matter how small, so a more technical definition of an earthquake is that an earthquake is the sudden slip of part of the Earth's crust, relative to another part, along a fault surface.
Of course, these answers don't begin to explain why, where, when or even how earthquakes occur. Instead, we're left with even more questions. Two of the most obvious to follow up with are:
"What causes this sudden slip in the Earth's crust?" and, "What, exactly, is a fault?"