Fading in or fading out are essentially increasing or lowering the volume gradually. While it can be done by just using the volume control, it is handled differently in most modern audio editors. Most often you decide how long the fade should be (i.e. how long it should take for it to reach the full volume), and in some cases also the fade curve. The curve decides the behavior of the fade, for instance if it perhaps should reach a high volume quickly and fade in the last bit slowly or vice versa.
A crossfade is a fade between two signals (“pieces of audio” if you will) that fades out from one and into the other. It can be used for cool transitions form one signal to another, but it is also very common to use as a editing tools for cleaning up noise.