The octave buttons are used to shift the keyboard up or down in one octave increments. This extends the range of the keyboard in either direction, which lets you play notes that lie outside the number of keys available.
The more buttons the device has, the more complex the input can be.
Faders can be either channel volume faders, pitch sliders, crossfaders or even EQs.
More knobs (rotating controls) can give the user more dedicated control over the device – e.g., volume, effects, EQs, etc. Types of knobs may differ between endless type or step type.
The pitch wheel alters the pitch and timbre of a sound in real-time. It can give fine tuning control from fractions of a semitone to huge multioctave pitch swings.
The mod wheel is used to add expression, or to modulate (change) various elements of a sound or sample.