What are they and what are they used for? And how are they different from regular SLR or DLSR cameras?
"Format" refers to the size of the recording medium in a camera. I say recording medium because the term originated in the film era and has continued to the digital age.
There are no hard limits but medium format is typically anything larger than 35mm film up to 6cmx7cm film. Large format is typically everything from 4"x5" up.
In addition to the size of the film or sensor there are a number of features typical of medium and large format cameras: