Beat
The pulse, struck at regular intervals, that you use to count musical time. Think of it as the place where you could clap along. Much dance music is four on the floor, so it sounds overlapped with the kick, but the beat and the kick are not the same thing. Even in a breakdown where the kick does not sound, the beat keeps flowing.
Counting beats
Play a track and tap your foot along to it. Each place your foot hits the floor is one beat. In dance music, four beats make one bar. You count "1, 2, 3, 4" and go back to "1."
The beat that falls on this "1" is called the downbeat. The downbeat is where the music feels heaviest, and changes in the track often start there, so it works as a landmark while you count.
Why DJs care about the beat
When you overlap two tracks, if their beats sound at the same instant, you hear a single rhythm. If they are even slightly off, the beats sound doubled, like a galloping horse: "ba-dum, ba-dum." Lining them up is called a beatmatch.
If you align not just the beats but the start of the bar, and the start of the phrase, a larger unit of the track, both tracks change at the same time and the mix sounds natural.
Beat vs. kick, beat vs. BPM
The kick is the "boom" of a drum sound; the beat is a mark that divides time. In a four-on-the-floor track the kick plays on every beat, so they look like the same thing, but when the kick drops out in a breakdown the beat keeps going. Not stopping your count when the kick disappears is the key to not losing your place in the mix.
BPM is a number for how many of those beats there are in a minute. Think of the beat as the marks on a ruler and BPM as how finely they are spaced.