There will be two sections here...
Adding to a new aquarium.
Adding to an aquarium with occupants in it already.
Adding fish to a new tank:
So far, let's hope you have tested and properly adjusted the pH and temperature within the tank. Now you have arrived home from that long trip to the fish store and your arms are all loaded down with water filled plastic baggies. Hopefully not too many, as you do not want to introduce too many fish into your tank at any one time! Doing so can cause an imbalance in the water as we discussed in part three of this series (see link below). For a good average, add no more than 3 inches of fish per 20 gallons of water in any seven-day period of time. This slow method allows for the biological portion of your filtration system keep up with the amount of fish (bio-load) of your tank.
For arguments sake, let's say you were smart and only brought home a few fish to start your tank off with. Since your tank is brand new, the best species to start a tank off with are White Cloud Mountain Minnows (Tanichthys albonubes), closely followed by species such as Danios and guppies.
Ideally, you would have a secondary tank called a hospital or quarantine tank to keep these fish in for about 14 days prior to adding to your community or show tank. A quarantine tank is usually just a small 5 or 10 gallon tank that allows you to isolate the fish for a couple of weeks to ensure your new additions are not carrying any diseases or parasites. Granted, I know a lot of first time fish keepers only have one tank, and do not have either the room or the finances for a second tank. It is not required that you have one of these secondary tanks, but as you get further immersed into the hobby you will find them invaluable. For the rest of this article, however, I will assume that you only have one tank.
Place the baggies of fish into your tank WITHOUT opening them. Let them rest floating in the water for 15-30 minutes. The fish will be fine and NOT run out of oxygen. What we are doing is making sure that the water temperature between the tank and baggies equalizes. We sure as heck don't need to put our new fishes into a state of shock!
After this wait you will want to open one bag and add some water from your tank into the baggie. In a few minutes add a bit more water, and so on. If you need to remove some water from the baggie, make sure this water DOES NOT enter your tank; discard it outside, onto plants or into an available sink. This whole process may take 20 minutes or more. This method is there to make your new fish feel comfortable in case there is (and often there is!) a pH difference between your tank and the water from the pet store.
Now here comes the place where you need to decide what you want to do: Many people take the water filled baggie out of the tank, hold it over a sink, and then pour out all the water and the fish into a fish net. Then take the netted fish and add them to the tank. This ensures that no "bad" water form the pet store enters your tank. While this method does have its merits, often times with barbed fish like Raphael's, Gourami's, Angelfish and the like, injuries from being caught up in the net can occur, so it pays to be extra careful.
Other people will simply open up the baggie in the tank and allow the new fish to swim out on their own when they are ready. This way no harm or undue stress can be inflicted upon the fish, but it does make your tank susceptible to infected water from the pet store. (Another reason to have the quarantine tank!) So whichever method you choose is up to you; neither is absolutely wrong or absolutely right.
Here is the easy part, and quite natural: For the next 10-30 minutes, just watch your fish. Look for any unusual or odd behaviors. Now it is quite common for a fish just introduced to a new environment to head straight to the bottom of a tank, or into some hiding place. As long as everything seems to be going along just fine feel free to follow the above steps in releasing the remainder of fish. Do not feed them right away as they are adjusting and need at least an hour or two to mark out their new territory.
Adding fish to an existing tank:
So your tank has been running well for a while now and you are ready to add some more critters to it.... If you have made any kind of an investment in your fish I HIGHLY recommend you set up the previously discussed quarantine tank (that smaller, secondary aquarium). You will want to keep your new fish separate from the main tank for approximately two weeks. This will allow you time to treat the fish in case of any disease. Disease is very common in newly acquired fish due to the stress placed on them during transportation.
Just prior to adding your fish to your community tank, do some house cleaning in it and move around some of the decorations, rockeries and the like. Introduce your new fish within an hour of doing this. Most fish are territorial and have set up there own "place" within the tank. New fish would either have no place to call their own or have to fight your existing fish for a spot in the tank. By rearranging your decorations/rockeries this allows for both new fish and old fish to claim a new part of the tank as their own. Keep an eye out for fighting. Some species just shouldn't be mixed. Be prepared to remove the offending fish if this occurs (yet just another reason for having the quarantine tank).
The biggest rule I really want to get across to everyone is NOT to add too many fish too quickly. All too often we have those impulse shopping urges and just "have to get" that additional cool fish or two. Just remember if you are walking up to the checkout stand and it takes two hands to hold all the fish, chances are you are going to meet up with some problems in the very near future. Besides if you only get three fish a week, it's a great excuse to revisit the fish store again next week to see what new additions they may have brought in.