Definition
The focus (also called the hypocenter) is the actual spot underground where an earthquake first begins, and the epicenter is the point on the surface directly above it. The focus has depth, since it sits inside the Earth, while the epicenter is the location on the ground that you see on a map.
The starting point underground and the dot on the map
Picture turning on a light bulb in a dark room. There's the bulb itself, and there's the spot on the ceiling where its light lands straight on. Think of the bulb as the place deep underground where rock breaks, and the ceiling as the ground you're standing on. Earthquakes don't happen magically at the surface. Most of them begin anywhere from a few miles to dozens of miles below ground.
The place where rock snaps and seismic waves first begin underground is called the focus. It has latitude and longitude, plus a depth. So it's a 3D location.
But when an emergency alert or the news says "an earthquake hit near such-and-such area" and drops a dot on a map, that's the epicenter. Draw a straight line from the focus up toward the surface, away from the Earth's center. The spot where that line meets the ground is the reference point on the surface.
Why shaking is usually strongest near the epicenter
Toss a pebble into a calm pond and ripples spread out in circles. The shock from the focus works the same way. It travels as seismic waves (vibrating energy released by an earthquake) and shakes the ground in every direction. The farther the waves go, the more their energy spreads out, so they get weaker.
The epicenter is the surface point closest to the focus. Someone standing there is nearer to the source than anyone else, so they generally feel the strongest shaking, and feel it first.
But how hard the ground actually shakes depends on more than distance. It also depends on the ground itself. Even if you're a bit away from the epicenter, if you're standing on soft ground like sand or mudflats instead of solid rock, the waves can get amplified and the shaking can be even worse.
To be more precise: it's a surface, not a point
In earthquake news, the focus and epicenter are each shown as a single dot on a map. To be more precise, though, a real earthquake isn't a single event at a single point. It's a huge fault surface slipping and breaking.
The focus is where a giant block of rock underground first gives way and begins to crack. That break doesn't stay put. It tears outward along the fault, like a rip spreading through fabric. In a big earthquake, it can run for dozens or even hundreds of miles in the blink of an eye.
That's why, in a major earthquake, damage can be severe not just at the epicenter but along the whole stretch the fault line passes through. Also, for the same magnitude, a shallower focus means the energy reaches the surface with less loss, so damage tends to be worse.
Common misconceptions
The earthquake starts at the epicenter.
An earthquake starts deep underground, at the focus. The epicenter is just the point on the surface directly above the focus.
Where you meet it
When the news says "6 miles south of a certain city," it's reporting the epicenter, a location on the surface.
When it says "it struck at a depth of 6 miles underground," it's telling you where the focus is.
The USGS gives both: a map location for the epicenter and a depth for the focus.
In one sentence
The focus is the real spot underground where an earthquake begins, and the epicenter is the point on the surface right above it.