When I started observing kashrut, the first question I had about eggs was whether they were considered to be milchig (dairy) or fleishig (meaty). They are produced by animals like milk, so that could be an argument for treating them like dairy. On the other hand they are "potential meat" in the sense that they could develop into a bird, so that could be an argument for treating them like meat. In the end, I learned that eggs are considered parve (neutral, neither milk nor meat) – which of course fits best!
So what makes an egg kosher? Simple, it has to come from a kosher bird (see the list of kosher birds). Most commercially sold eggs in Western countries are from chicken (or they are labeled as non-chicken and much more expensive if not), so there shouldn’t be too much of a problem.
The only complication is that eggs may contain blood spots. If you crack open an egg and find blood, you should discard the egg. It is a good practice to open the egg into a separate bowl before putting it with the rest of the eggs/dough/whatever you are making to avoid having to throw out the other food if one egg has blood. Boiled eggs need not be checked, but some have the custom to discard the egg when they see a red spot during peeling. Kashrut agencies and kashrut books contain detailed advice about what sort of spots eggs can contain and in which cases you should discard them (e.g., Eggs and Blood Spots by the OU).
This should cover everything for the home-use of eggs. Industrial egg products are a very different story and may require supervision (if you really want to know why… R. Zushe Blech: Industrial Eggs – Not As Simple As It May Seem).