The event horizon of a black hole is the boundary (‘horizon’) between its ‘outside’ and its ‘inside’; those outside cannot know anything about things (‘events’) which happen inside.
What an event horizon is – its behavior – is described by applying the equations of Einstein’s theory of General Relativity (GR); as of today, the theoretical predictions concerning event horizons can be tested in only very limited ways. Why? Because we don’t have any black holes we can study up close and personal (so to speak) … which is perhaps a very good thing!
If the black hole is not rotating, its event horizon has the shape of a sphere; it’s like a 2D surface over a 3D ball. Except, not quite; GR is a theory about spacetime, and contains many counter-intuitive aspects. For example, if you fall freely into a black hole (one sufficiently massive that tidal forces don’t rip you to pieces and smear you into a plastic-wrap thin layer of goo, a supermassive black hole for example), you won’t notice a thing as you pass through the event horizon … and that’s because it’s not the event horizon to you! In other words, the location of the event horizon of a black hole depends upon who is doing the observing (that word ‘relativity’ really does some heavy lifting, if you’ll excuse the pun), and as you fall (freely) into a black hole, the event horizon is always ahead of you.
You’ll often read that the event horizon is where the escape velocity is c, the speed of light; that’s a not-too-bad description, but it’s better to say that the path of any ray of light, inside the event horizon, can never make it beyond that horizon.
If you watch – from afar! – something fall into a black hole, you’ll see that it gets closer and closer, and light from it gets redder and redder (increasingly redshifted), but it never actually reaches the event horizon. And that’s the closest we’ve come to testing the theoretical predictions of event horizons; we see stuff – mass ripped from the normal star in a binary, say – heading down into its massive companion, but we never see any sign of it hitting anything (like a solid surface). In the next decade or so it might be possible to study event horizons much more closely, by imaging SgrA* (the supermassive black hole – SMBH – at the center of our galaxy), or the SMBH in M87, with extremely high resolution.
The Universe Today article Black Hole Event Horizon Measured is about just this kind of black hole-normal star binary, Black Hole Flares as it Gobbles Matter is about observations of matter falling into a SMBH, and Maximizing Survival Time Inside the Event Horizon of a Black Hole describes some of the weird things about event horizons.