A. Compact cameras, sometimes also called point-and-shoot probably were your first camera. They are very convenient: cheap, small, light and fool proof. As the name suggests, just point it in the general direction of the subject and press the button. The camera does the rest.
Their main advantages, as said, is their low profile. They are so small and unobtrusive that Blog you are likely to carry them all the time, and to have them handy when you need them. After all, even the crappiest camera you have with you beats the amazing one you left at home. Their small size is also an advantage when you want to be discreet. Most people will assume you are just a tourist and won’t give you a second look, whereas even a small DSLR will attract attention.
B. Digital Single Lens Reflex cameras (DSLRs) are the “serious” camera of choice these days. Though this comes at the price of a serious increase in weight and bulk (and, well, price), they are also much more uncomprimising on everything that matters. In particular, they have interchangeable lenses which allows you to always have the best lens for the occasion. Even APS-C (DX) cameras have big enough sensors to allow shallow depth of field and good low light/dynamic range quality. There is an optical viewfinder, which allows framing in the worst light conditions and is generally more responsive than any electronic screen.
The annoyances of compact cameras are also gone: shutter lag is virtually unknown, autofocus generally very fast (though this depends on the lens) and even entry-level cameras provide full manual control along with their scene modes.
C. Mirrorless (or EVIL, for Electronic Viewfinder, Interchangeable Lenses) cameras are new hybrids which started appearing in 2008. There are different standards: Sony has NEX, Panasonic and Olympus use micro-4/3 and Fuji has the X-series. The concept is to remove the bulky mirror and pentaprism necessary for the optical viewfinder of a DSLR, but to keep the other capabilities, in particular large sensors and interchangeable lenses. This allows for a drastic reduction in size, putting them closer to compacts than DSLRs. Whether the sacrifice of the optical viewfinder in exchange for a smaller size is worthwhile will be an entirely personal choice.
D. The big stuff refers to bigger than 35mm cameras, which in the digital world means medium format backs. The cheapest start at 10-15k$, without lenses, but their resolution and image quality is hard to beat. They have little interest if you are not printing big, as the difference from high-end DSLRs will be hardly noticeable. They are mostly used by commercial shooters and (rich) landscape photographers.