Continuum mechanics is a branch where the material is considered to have continuous distribution of constitutive properties (material properties) and the whole modeling scheme is typically performed without intrinsic length scales (with some exceptions of course, but I'm thinking you're interested about the elementary end). Discrete is the reality material wise.
Yea - what PII said. You treat the material as if you didn't know it was composed of discrete (i.e separate) atoms.
The remarkable thing about continuum mechanics is the range of size scales over which it works - you can use the same mechanics to describe the behaviour of a few drops of water or a whole ocean.
Just adding or supplementing to whatever has been said... A continuum is basically the smallest possible chunk of stuff you can analyze without having to take atoms and molecules into account. Because the stuff in the universe cannot be cut in half an infinite number of times. We eventually get to a point when we're at the atomic level. So continuum mechanics works as long as you make the continuum large enough to neglect the quatumn physics involved in atomic level interactions. That being said, you can imagine the size of the contiuum for different materials will be different since they are made of different stuff in the molecular level.