There are a number of things that can cause dead fish. First, you can overfeed them. If you feed them before they are hungry (about halfway down on their "Fed" meter), they will lose health each time you feed them.
Fish can also lose health and die if they are too fragile to survive in the level of Fish Environment that you currently have. Only the common fish (the ones that you can hatch from common eggs) can remain completely healthy in a Level 1 Fish Environment. The very fragile hybrids will usually not live to adulthood. Vitamins will temporarily improve their health, but only the proper Fish Environment will solve the problem.
Of course, sick fish will die if you don't cure them, and diseased fish can infect other fish, compounding the problem!