Sound synthesis techniques

The most important goal of sound synthesis is to produce sounds which sounds like real musical instruments. The common method for sound synthesis is by using oscillators. Oscillators produce desired waveforms by mathematical computation. A higher quality reproduction of a natural instrument can typically be achieved using more oscillators, but increased computational power and human programming is required

Amplitude envelope
One of the salient aspects of any sound is its amplitude envelope. This envelope determines whether the sound is percussive, like a snare drum, or continuous, like a violin string. The envelope defines how the amplitude of the time varies with time. A sound’s amplitude profile is described by "ADSR" (Attack Decay Sustain Release) envelope model.

Attack time is the time taken for initial run-up of the sound level from nil to 100%.

Decay time is the time taken for the subsequent run down from 100% to the designated Sustain level.

Sustain level, the third stage, is the steady volume produced when a key is held down.

Release time is the time taken for the sound to decay from the Sustain level to nil when the key is released. If a key is released during the Attack or Decay stage, the Sustain phase is usually skipped. Similarly, a Sustain level of zero will produce a more-or-less piano-like (or percussive) envelope, with no continuous steady level, even when a key is held. Exponential rates are commonly used because they closely model real physical vibrations, which usually rise or decay exponentially.

Synthesis methods
There are also many different kinds of synthesis methods, each applicable to both analog and digital synthesizers. These techniques tend to be mathematically related, especially frequency modulation and phase modulation. You can find the details of these algorithms in any book dealing with Digital signal processing.

Subtractive synthesis
Additive synthesis
Granular synthesis
Wavetable synthesis
Frequency modulation synthesis
Digital sampling