"The last few cans of fish food I bought said Do Not Overfeed. Why is that a problem, and how much should I feed my fish?"
Overfeeding is the most common mistake fish owners make. Overfeeding clogs the filter, and breaks down into toxins that are harmful to fish. Hence the warnings on the packages.
In nature fish eat whenever they are hungry and food is available. If food sources are plentiful, they will eat several times a day. On the other hand, if food sources are scarce, they might go for days between meals. For this reason, fish are very opportunistic and will eat whenever they have a chance. That means that if you offer them food, they will usually gobble it up even if they aren't starving. Keep that in mind the next time your fish 'beg' for food. Fish quickly learn who brings the food to the tank and will jump at the chance to be fed, even if they are not in dire need of food.
So how often, and how much should fish be fed? Frequency will vary based on the type of fish. In general most fish do quite well on one feeding per day. However, some owners prefer to feed their fish twice a day. Regardless of one or two feedings, the key is to keep the feedings very small. The timing is not critical, with the exception of nocturnal feeders, such as certain catfish. If you have nocturnal fish in your tank, be sure to feed them just before turning the lights out at night.
There are some exceptions to the once per day feeding rule. Herbivores (vegetarians) need to eat frequently because they do not have large stomachs to hold a lot of food. In nature they would graze all day long on plants. They should be given several small feedings a day, or provided with live plants they can nibble. Newly hatched fry and young fish not fully grown, require more frequent feedings of special foods designed for fry.
As for how much food to feed, a good rule of thumb is to feed no more than the fish will consume completely in less than five minutes. When in doubt, underfeed! You can always give them another small feeding if necessary. However, if you overfeed the uneaten food will produce by-products which can be harmful to the fish. In the event you do overfeed, promptly remove the uneaten food using a siphon or net.
As a final note, keep in mind that not only the amount but the type of food is important. Below is a link to a handy table that shows the proper type of diet for most popular aquarium fish.
Dietary Types - Popular Aquarium Fish