What is the difference between 1/3, 1/2, and full casters?

I saw in a recent answer a reference to 1/3 casters and 1/2 casters, and have never encountered this terminology before:

Paladin’s are known as 1/2 casters, so their spell progression isn’t as big as full casters.

What does it mean to be a 1/3 caster vs a 1/2 caster vs a full caster? How does one distinguish between the three, and which classes correspond to which types of caster?