Answer: Drone is the simplest - a tone or tones held while other parts play. Think of the drone pipes on a bagpipe: they sound one or two pitches continuously while the melody pipe handles the tune. This is an ancient device and a very effective one. Typically a drone will be on the tonic pitch - the first note of the scale, or perhaps the tonic plus the dominant (fifth note of the scale), sometimes combined with one or more octaves of those pitches. A drone in classical music is often used to suggest a rural scene with peasant dancing, etc.

An ostinato (derives from the Italian for "obstinate") is a short melody, perhaps one or two measures, that repeats over and over while other parts play. This is also an old device that goes back at least to the 13th century, and today is heard very frequently in popular music - there must be thousands of rock or blues songs that use ostinato. Let me pick an oldie: "Smokestack Lightning" by Howlin' Wolf:

There's a little more on ostinato and related terms chaconne and passacaglia in Chapter XII of Exploring Theory with Practica Musica.