What you'll need
Here are some basics to keeping cardinal tetras the right way and saving yourself and the tetras lots of grief in the future.
Tank
You really should have a 10 gallon or larger tank, but you can get away with something like an 8 gallon biorb or 9 gallon biube (approx 35 liters). Tetras are beautiful when they school, and this is necessary for them to show most of their colors (schooling is a defense mechanism for fish) so you want at least 6 of them. A 5 gallon tank is really pushing it, but if you were VERY diligent about the water parameters, it's possible.
Water Parameters
The pH of your water should be ~ 5.5 to 7.5. Whatever it is, try to keep the fluctuations to an absolute minimum. If you don't currently test for pH, now is a good time to start. It's more important for cardinal and neon tetras than it is for quite a few other popular freshwater fish. You also should keep nitrates to a minimum with regular water changes and/or live plants.
Heater
Cardinal tetras require a temperature of between 73-81°, but the important thing here is to minimize as much as possible the fluctuations in temperature. That generally requires an aquarium heater. A stealth heater (50 watt for a 10 gallon, 100 watt for a 20 gallon, and so on) is ideal.
Filter
You really need a filter for ANY fish, despite the fact people keep bettas and goldfish in unfiltered tanks or bowls. But in this particular case, it is a must have. You simply cannot have a tropical fish tank without a filter.
Lighting
With cardinal tetras, you really want at least some part of your tank with subdued lighting. They'll be fine with just a standard issue light from an aquarium kit's hood, but you really want to give them cover somewhere in the aquarium.
Water Conditioner
With cardinal tetras, you have to be diligent about the water parameters. Pick up Stress Coat or Prime- these are two extremely popular water conditioners that will get rid of chlorine so your water is safe to add to the fish tank.
Substrate
You need gravel or some type of substrate to line the bottom of the fish tank. Cardinal tetras aren't picky here, but if you're adding loaches or any type of catfish as a tankmate, you should stay away from rough ceramic media as this will scratch their faces. If you are adding live plants to the tank, then this is slightly more complicated in what substrate would be best. Standard issue aquarium gravel is fine otherwise.
Testing Kit
You will need the ability to test for ammonia, nitrites, and nitrates. Many people just use test strips, and these are fine for a guideline, but they're often not the most accurate thing in the world. Aquarium Pharmaceuticals makes a freshwater master test kit (which would also let you test pH), so that is something you may want to consider. No testing supplies available to hobbyists are 100% accurate...ammonia especially is a difficult test, so keep this in mind as well when taking readings. One other thing...many local fish stores and some chain pet stores will do a free water test, so you may want to confirm some readings with them.
Optional Items
You will probably want to decorate your tank, so any decor you buy, just make sure you rinse it off before putting it in the tank, as you don't want the water to get dusty. You might also consider a bacteria booster for starting the nitrogen cycle, though opinions are mixed on the effectiveness of these.