Wikipedia defines sound effect as artificially created or enhanced sounds, or sound processes used to emphasize artistic or other content of the media.
Audio effect device is a piece of hardware or software that alters how the audio sounds. Digital audio is processed by a software that is usually implemented as a DAW plug-ins and allows combining many kinds of effects to get new unique sounds.
Sound effects can be classified in three general categories: dynamics, time based effects and filters.
Dynamics aka amplitude effects are used to dynamically control the amplitude of the sound and change the volume based on the material over time.
Compressor reduces dynamic range of the audio signal and rises above certain threshold level. The sound become flatter, it allows to increase the loudness of the signal or stabilize the volume.
Limiter limits the signal level above certain threshold. This is the extreme form of compression.
Expander increases the dynamic range of the signal and creates greater range between quiet and loud signal.
Noise gate amplifies the sound by eliminating the signal below a certain level in order to cut the noise off.
Time based / delay effects
Time based effects control propagation qualities of the sound and give the sense of the space. Audio signal is repeated and mixed with original sound using different delay time.
Delay is the basis of other time based effects. Original signal is followed by delayed repeat.
Repeater takes small portion of the sound and keeps repeating it.
Reverberation consists in many sound reflections during short period of time and creates acoustic space.
Chorus adds multiple short delays of variable length and mixes with original sound. It produces an effect of multiple similar sources and makes the sound warmer.
Phaser alters the signal by creating series of peaks and troughs in the frequency spectrum and mixing the result with original sound.
Flanger is kind of phaser produced by mixing of two identical signals together, with one signal delayed by a small and gradually changing period of time.
Filters manipulate sound's frequency and control timbre.
High pass filter passes high-frequency signals and reduces the amplitude of signals with frequencies lower than the cutoff frequency.
Low pass filter passes low-frequency signals and reduces the amplitude of signals with frequencies higher than the cutoff frequency.
Band pass filter passes frequencies within a certain range and rejects frequencies outside this range.
Band stop filter rejects frequencies within a certain range and passes all other frequencies outside that range.
Other equalizers: parametric EQ, graphic EQ, etc.
Other classes of sound effects
Distortion creates overtones by amplifying the audio signal until it gets too load and it changes the waveform shape.
Phasers, flangers, choruses, ring modulators, tremolo and vibrato can be classified as modulation effects and based on multiplying and altering the signal.
Pitch-shifters and harmonizers are based on changing the pitch or adding new frequencies to create harmony.