Mayonnaise is something that is incredibly easy to make and is delicious when it is homemade. It really doesn't even compare to the store-bought variety at all. The biggest drawback is that it doesn't keep very long. Homemade mayonnaise is a great base for other things as well - rouille, aioli, and so on. I sometimes add other flavorings as well - I'll add lemon if we'll be eating it with seafood, for example.
The only real ingredients you need are olive oil and one or more eggs, depending on how much you want to make. One egg makes quite a bit of mayonnaise - about a cup or so - so if you don't know how much you are going to want, start slow and then make more if you need it.
The one piece of equipment you need if you want to make mayonnaise easily is a blender with a lid that you can remove while it is running. It's even better if you can remove only a small part of the lid (such as a cap in the center) to avoid making a big mess. This is not totally necessary - mayonnaise is traditionally made by hand with a whisk, in which case you need to add the oil very, very slowly - I don't have much luck with this method, as I usually don't have the time or the patience.
Break the egg into the blender. Turn the blender to the lowest setting. Add the olive oil in a slow stream, until the mixture emulsifies. This will take a surprising amount of olive oil. I use good, expensive olive oil, because I want a good, quality mayonnaise. Keep in mind that the mayonnaise will take on the flavor of whatever oil you are using. You will hear a difference in the blender as the mixture emulsifies. Continue to add oil and blend if you would like a thicker mayonnaise, otherwise, stop when you're happy with the consistency. Refrigerating it also thickens it up a bit. Add a tiny bit of salt, to taste, but this is optional.
To make aioli, add some garlic with the egg in the beginning. Go easy on the garlic. Probably less than one clove per egg. If you're using more than one egg, try making just one batch first, so you can taste the pungency of the mayo and adjust for the second batch.
Rouille is a type of spicy French mayonnaise, which I use for soupe de poisson. While there are many recipes on the internet, and if you ask people from Southern France, you will undoubtedly get numerous different recipes. I found it easiest to make an aioli base, then stir in some powdered cayenne pepper, and saffron (optional). I serve it on croutons or crostini, dunked in the soupe, with gruyere or other swiss cheese.
Final note: Sometimes the mayo will "break", meaning it either doesn't emulsify, or it starts to emulsify and then separates again. If you are struggling with this, it could be that you are adding oil too quickly. Remove the entire mixture from the blender, but don't throw it away! Start again with another egg, then add your broken mixture as if it were just oil, very slowly. You'll end up with a double batch, but I'm sure you can think of recipes for which to use it!