Synthesis exploits the fact that both electricity and sound have frequencies. If you run an electric signal at a certain frequency into a speaker, a sound with that frequency will be played. There are two kinds of synthesis: additive synthesis, where harmonics are added to a tone, and subtractive synthesis, where harmonics are subtracted from a tone. These two methods can be combined in a wide variety of ways to generate any sound imaginable, and synthesized tones often use both to create exactly the right effect.
All synthesis begins with the oscillator. An oscillator generates a frequency for the synthesizer to use, setting the pitch of the sound that will be generated as well as the waveform of the sound. Higher frequencies result in higher tones, and lower frequencies result in bass tones. A good synthesizer will have two or more oscillators for generating as many different kinds of frequencies as possible, and can even create frequencies outside the range of human hearing. This frequency will then move to the filter.
When the frequency is run through the filter, the filter will remove certain frequencies and harmonics in order to better replicate the sound you are trying to synthesize. Filters are dynamic, meaning that as you change the pitch or other aspects of the sound, the filter will shift what frequencies it is removing in order to continue replicating that instrument properly. Some frequencies may have to be run through several different kinds of filters in order to achieve the proper effect.
The amplifier is what takes the voltage you have been generating and gives it the power needed to turn it into sound. Synthesizers use small amounts of electricity to create the frequencies needed, and those cannot be heard over most sound systems due to their low power. The amplifier will increase the power, expressed as amplitude in the wave you are generating, of that signal and make it powerful enough to be heard over speakers.
The envelope generator is the part of the system that decides when sounds start and end. It can have a wide variety of effects and can be applied to the oscillator, the filter or the amplifier. When applied to the oscillator, it can change the pitch; with the filter, it can change the brightness of the sound; and with the amplifier, it can control volume. It also controls timbre for all three.