Sidechaining

Sidechaining is where an audio signal is used to trigger an effect, it is most commonly used to trigger a compressor, with the kick drum as the trigger. As a result when somebody says sidechain without any other context it is most likely they are refering to sidechain compression.

Sidechain Compression
Sidechain Compression refers to the compression of an audio-signal, triggered by second audio-signal, often the kickdrum. This gives the mix a sense of drive, as if it is pumping, as audio is swelling in and out around the kick.

Examples of Sidechain Compression
DJ Mehdi - Signatune (Thomas Bangalter Edit)

Eric Prydz - Call On Me

Use of Sidechain Compression with other effects
Reverb

White Noize

Sidechain EQ'ing
Also known as 'Smart Mixing' Sidechain EQ is an EQ that is triggered by the audio output of another piece of audio. Mostly used in DJ'ing, Sidechain EQ makes areas of an audio signal quieter depending on how much the 'trigger' audio signal is outputting, making room for the 'trigger' audio signal.

Sidechain EQ'ing is infact done with compressors, rather than an actual EQ device, tracks are split into three (or more if desired) frequency bands, created by EQ's, which are sent to Bus channels, which are then used as the sidechain trigger.

Gate Triggering
Gate triggering is where one audio-signal is used to open the gate of another channel.