single shot shotguns are generally the least expensive; their downfall is that they only hold one shell at a time. To fire again, you have to open the breach and insert a new shell, a time consuming activity. Still,
they are a functional tool for many tasks.
Pump-action shotguns are the most common kind. They are known for their low cost (though some models can still get very expensive, they are some of the cheapest common shotguns), and reliability. After every shot you must pump the forend (place where you put your left hand) of the gun to eject the spent shell and load another from the magazine tube into the chamber.
Auto-loader shotguns (also known as "semi-automatic") use a few different types of systems so the only operation you do to fire a round is pull the trigger. Reliability is lowered and using a variety of rounds reduces reliability even more.
Double barrel shotguns come in two varieties: over-unders (O/U) with one barrel above another, and side-by-sides, in which the barrels are located horizontally. Some people prefer one style, some the other; neither is ultimately better, and both styles are expensive.
For a beginner, a pump action is probably a good place to start since it is very common, cheap, versatile, and reliable.