I'll list off the various necessities. All of these are essential to fish health.
The nitrogen cycle
The most important and immediate issue in fishkeeping is the nitrogen cycle. The nitrogen cycle is the process that removes fish waste from the water.
Basically, fish waste breaks down into ammonia (which is HIGHLY TOXIC and KILLS FISH)
Ammonia is broken down by bacteria into nitrite (which is also HIGHLY TOXIC and also KILLS FISH)
Nitrite is broken down into nitrate (which is relatively harmless at reasonable levels) Nitrate is eventually removed in water changes or by plants.
The problem here is that the bacteria necessary to do this aren't immediately in your aquarium in large enough numbers. They need to multiply before they can handle the amount of ammonia and nitrite produced. The bacteria that convert these toxic chemicals tend to double in number every day, so, at the end of a month, 1 bacteria has become about 4 billion. It tends to take 2-4 weeks to complete the process of stabilizing your ammonia and nitrite levels at zero. Until this is achieved, the spikes of ammonia and nitrite tend to look like this:
These spikes cause much frustration in new aquarium owners, because they tend to kill fish.
To start this process off, you need "food" for the bacteria. You can do this with fish, in which case you need to slowly increase the number of fish over a few months until you reach your full stock, or you can add an ammonia solution with no detergents or fragrances. They sell this at ACE hardware (in a 10% solution). Start off with 5 drops of 10% solution in 20 gallons, test for ammonia (I recommend getting a proper test kit with a test tube and a bottle of test solution, available at any aquarium store) and you should have about 3 ppm. Test daily, and keep it at about 3 ppm. If you do too much, you can do a water change (the bacteria colonize surfaces, they're not in your water.) Always use dechlorinated water for water changes, as chlorine will kill your beneficial bacteria.
There's a couple ways you can speed the process up:
You can buy the bacteria in bottles. There are several companies that sell this. The most respected one is probably seachem (their bacteria product is called stability.) This won't completely establish your bacteria immediately, but it should shorten the time to complete this process down to a week or less.
You can also use filter media taken from an established tank to "seed" a new tank with bacteria.
Live plants help immensely because they come with some free bacteria and they eat ammonia themselves. Make sure you have appropriate lighting. If you heavily plant the tank, you can probably just start adding fish.
Grows best at 86F, so crank that heater if you're using the pure ammonia solution.
Once your tank is cycled, it's ready to be fully stocked with fish.
Heating
Tropical fish require 74-82 degrees fahrenheit. A submersible aquarium heater accomplishes this.
In a 20 gallon tank, to increase the temperature 9 degrees above room temperature you'll need a 50 watt heater. If you need more than that, go up to a 75 or 100.
Filtration
Filtration accomplishes 3 things:
Mechanical filtration removes particulate matter from the water, making it look clear.
Biological filtration is housing for the beneficial bacteria I talked about earlier. Basically, it's just some material with a LOT of surface area for the bacteria to colonize. You never ever want to clean your biological filtration. If it gets really clogged up, swirl it in a bucket of tank water (NEVER tap water.)
Chemical filtration is any kind of chemically active media. You can get all sorts of stuff. The most common is carbon, which removes substances from the water via adsorption. I personally don't use carbon. You can skip this step entirely if you want.
You want a filter that has a lot of flow. 5-10x turnover rate per hour is good, which is 100-200 gallons per hour in a 20 gallon. Types of filters include: air powered filters (I don't recommend using these, because they don't have good mechanical filtration), hang-on-back power filters (most common), canister filters (least visible, most customizable as far as media goes). You need flow so that fish waste will get pushed into the filter.
Stocking
There are lots of completely bunk rules for stocking. An example is: "1 inch of fish per gallon of water." The problem with these is they don't take into consideration the fact that the mass of the fish goes up exponentially with the length. So here I am putting my 6 foot (72 inches = 72 gallons, right?) shark into a 75 gallon tank (75 gallon tanks are 4 feet long.)
So how do you solve this? Simple: use common sense. Does it look crowded, or will it be crowded when the fish are fully grown? Yes? Less fish or bigger tank. No? Good! Carry on! Or, when in doubt, snap a picture, list your fish, and ask a forum.