An oscillator, in brief is something that oscillates, as per Webster's definition of oscillate:
1.a : to swing backward and forward like a pendulum b : to move or travel back and forth between two points
2 : to vary between opposing beliefs, feelings, or theories
3 : to vary above and below a mean value

In my experience, however, it most often refers to some sort of electronic signal generator. The signal might be a square wave, sine wave, triangle wave, or other periodic signal. There are many ways to make an oscillator, such as crystals or the 555 chip and its relatives. In many cases, the oscillator involves the charging and discharging of capacitors to produce a given voltage. Alternately, with a PIC chip or other microprocessor, one might just twiddle an output bit. This will directly produce square and rectangular waves, and with other components to produce waveform deformations, other waveforms may be produced.

Electronic oscillators are used for many things:

Sources: A recent Merriam Webster's Dictionary along with the electrical engineering classes I took in college.