An oscillator is something that oscillates back and forth, making a repeating waveform. How often it repeats is known as its frequency, measured in cycles per second, or Hertz. As we perceive this frequency as a note's pitch, synthesisers use oscillators to make all the pitched sounds, for pitched notes. This is in contrast to noise generators.
A synthesiser's oscillators can be voltage controlled, digitally controlled, or fully digital. It may also have a sub-oscillator.
A synthesiser's oscillator should have at least a single input, to tell it which frequency it should oscillate at. (Whether that input is physically visible and open to you depends on whether your synthesiser is modular or not.) A good oscillator should also have at least one more input which is mixed in with the main one, so you can introduce effects like vibrato. (The multiple CV inputs are summed together, as the oscillator effectively has its own built-in mixer.)
Most analogue oscillators (whether VCOs or DCOs) can output pulse, triangle, and sawtooth waves. More advanced ones might even attempt to output a sine wave, with varying degrees of success. You didn't hear much music with sine waves in until digital oscillators and samplers became popular.
A square wave is just a pulse wave that happens to be up half the time and down the other half. In other words, it has a duty cycle of 50%. Most oscillators let you choose the width of the pulse wave so it doesn't have to be square, and usually let you modulate the pulse's width with another input. Especially fancy oscillators can adjust not just the pulse's width, but also the triangle/sawtooth's width, smoothly transitioning between the shapes. (The Putney's oscillators are particularly versatile in this regard.)
Some oscillators, such as on the System-100M's 110 and 112 modules, only output one waveform at a time, with a switch determining its shape; others, such as on the 900 Series's 901 and 921 modules, output all the different waveforms simultaneously, ready to all be plugged into a mixer together.
Twice as nice
With two oscillators, you can detune them slightly from one another, to make a richer sound. If you detune them only very slightly, you might notice a slowly repeating sound. This is called beating, and it's a very similar effect to pulsewidth modulation. The closer together the two oscillators' pitches are, the slower the beating will be.
You can do more with two oscillators than simply mix their outputs together: you can make them directly influence each other, using oscillator sync, or frequency modulation. You can also combine their outputs using ring modulation, which is essentially amplitude modulation. So it's generally preferable to have a pair of oscillators if you can.
Personally, I haven't found a use for more than two oscillators per voice. As I generally only mix three or more together without them affecting each other, I can achieve a similar effect by simply recording the same part twice. With the analogue synthesiser's subtle imperfections, this noticeably doubles the entire signal path, not just the oscillators.