An electrical circuit which produces a cyclic (alternating current) signal, which at at any given time has a definite frequency. Oscillator circuits in general divide into two groups. The first group relies on the natural tendency of some component or combination of components to resonate; these include crystal and inductor-capacitor (LC) circuits. Resonant oscillators are easy to build, but it is very difficult to make their frequency voltage controlled. For this reason, most analog oscillators used in synths fall into the second category, which electrical engineers refer to as "relaxation" oscillators. A relaxation oscillator consists of two basic components, a mechanism that causes the signal to change in some fashion, and a "reset" mechanism which forces it back to the start of the cycle. Two basic relaxation oscillator circuits, the sawtooth core and the triangle core, serve as the basis for nearly all analog voltage controlled oscillator designs used in synths. Low frequency oscillators are also usually constructed using the same basic circuits as audio VCOs; even though not all LFOs are voltage controlled, it is usually convenient to use the same design methods and components for them.
(One other type of VCO, often seen in radio applications, is based on a a component known as a "varactor". This is basically a capacitor whose capacitance can be voltage controlled, and varactor oscillators and filters are often used as tuning circuits in radio transmitters and receivers. However, varactor VCOs seldom had a range of more than one octave, which makes them unsuitable for use in synthesizers.)
Digital oscillators work on different principles. A digital oscillator may employ the math of the desired waveform (e.g., the mathematical sine function) to compute a waveform on the fly, but more commonly, it simply reads a pre-computed waveform out of memory. This is known as a wavetable oscillator, and it has the advantage that a huge number of waveforms can be made available, if the oscillator can access the memory in which it resides. A digital oscillator requires, at some point, a D/A converter that can convert the digital information into an analog signal that can actually be fed to an amp and loudspeaker, and heard.
A hybrid technique is digitally controlled oscillator, commonly seen in the 1970s and '80s. These divide into two basic categories. One is basically an analog VCO which replaces the reset mechanism with a digitally timed and controlled mechanism. The other relies on generating a set of square waves at harmonic intervals using a digital timer circuit, and then combining them to produce desired waveforms using the mathematical principle of Walsh functions.