In video work, a transition is a effect that is usually used in the beginning, end or the change of a scene. Without the transitions, the scenes would just change instantly, and sometimes this is not desired, because it doesn't fit that well to the scene. These are usually used to make it clearer that the next scene is taking place in different place or at different time.

Typical and very common transitions include

Fade out, fade in
Gradual complete tone change to/from black, or (rarely) to some other color. Often, a scene ends in fade out, and next scene begins with a fade in. On audio side, the fades are often used similarly; sound fades out, and then fades in.
crossfade
The previous scene fades away, and next scene fades into the view at the same time - and soundtrack is also mixed together, or may fade just like in fade out/in to black.
wipe/sweep
The old scene is replaced gradually with the new scene, or the picture is gradually blackened and the new scene is revealed similarly gradually; the scene is "swept" away. Usually sweeps run from one side to another in diagonal or straight angle, but there are other kinds of sweeps - shrinking or growing circles, stars, hearts; clock-like sweeps, spirals, 3D sweeps (e.g. picture flips over, new scene on the backside) and so on...
push/slide
Like sweep, but the old picture itself moves away and reveals the new scene (while in sweep both scenes stay relatively on same place).

Most NLE video programs handle transitions like this with ease.

In case of film, the transitions can likewise be done digitally, using different kinds of exposure to produce a copy of the negative, or physically by adding transparent tape on top of the film. There are other methods; For example, in ancient days, fades to black could be done by gradually dissolving the film emulsion from copy of the negative using some kind of dissolvant (acid?).

Thanks to QXZ for alternate terminology.