This was brought up in the welcome forum, and it's a very important topic, so here it goes.
I've put together a list of things off the top of my head, in no particular order, and that's just the tip of the iceberg.
1. Trying to alter the pH, hardness or some other natural elements of your tap water. While these things can be done, it usually leads to huge swings, which is very hard on your fish.
2. Not adjusting the temp of your water when doing a water change. It needs to be as close as possible to the water in the tank.
3. Not putting newly acquired fish in a quarantine tank. Even the highest quality fish can bring new diseases or ailments into your tank.
4. Not adding dechlorinator to the tank (if you have municipal water).
5. Sticking contaminated hands into the tank. This can include a broad variety of things on your hands; including hand lotion, soap, nicotine, or many other toxins.
6. Adding too big a fish load. Even though they are small when you purchase them, you have to look at the adult size before bringing them into your home.
7. Over cleaning a newly set up tank. Bacteria needs a chance to build up on all of the tank surfaces, so if you keep scrubbing it down, it's not going to happen.
8. Over feeding. Add only as much as your fish can eat in a 5 minute period. Additional food is just adding more of a load to your tank.
9. Putting incompatible fish together. The tiny glass cube we keep our fish in are only a fraction of the space they have in the wild, and if they have issues, there's no where to run.
10. Exposure to airborne toxins. If someone sprays insecticides, cleaning solutions, or anything toxic, it can enter your tank and kill the inhabitants in an instant.
The nitrogen cycle is a whole issue in itself, so here a link about that. Understanding the nitrogen cycle
I hope that everyone else will contribute to this list, and help make all of our tanks as fish/plant friendly as we can.