What is exactly a Dobsonian, and what are the differences when compared with Schmidt-Cassegrain or Newtonian?

Dobsonian is a type of mount, whereas Schmidt-Cassegrain and Newtonian are different configurations of the optics inside the telescope. Dobsonian mounted telescopes are almost always Newtonian due to the Newt's ideal placement of the eyepiece.

This is a Dobsonian mounted Newtonian scope:

A Dobsonian mount does not track the stars as they move in the sky, so it's not suitable for some applications but it's a great "looking around" mount and is really easy to use. I've set people at my Dob before and they intuitively get it within a few minutes. My German equatorial on the other hand takes a good deal of expertise to use.

Dobs are great for large or heavy scopes because they effectively balance the scope so that moving the scope requires very little effort.

Technically, a Dobsonian is a type of mount: an altazimuth (up/down, left/right) mount with large bearings and no slow motion controls or locks. The large bearings make its operation smooth and, as a result, it has no need for slow motion controls or locks.

Any type of telescope can be mounted on a Dobsonian mount, but most commonly it supports large Newtonian reflectors, with apertures from 6- to 40-inches. Because it doesn't need counterweights, it is very light in weight for its size, so that even the largest Dobsonian reflectors are portable.