Carbon dioxide injection certainly is not necessary for all planted aquaria. You can have a beautiful planted aquarium without CO2, most likely if you limit yourself to a slow-growth setup, with less demanding plants and less fertilization. However, if you want to grow a wide variety of plants and have vigorous growth, you need higher light levels, adequate nutrients, and supplemental CO2.
Here is how a cheap Do-It-Yourself Yeast-Generated CO2 System can be constructed.
Drill a hole in the middle of the cap of a 2-liter softdrink bottle* slightly smaller than the diameter of a 3/16" air hose tube. Insert the tubing through the hole so that about 2 cm (or 1") is inside the bottle when re-capped. Cut the tube at a 45-degree angle (see picture) so that any water that gets onto the tube will drip back down more easily. Seal the insertion point with silicone caulk on both sides of the cap (the inside seal will be more important, due to the CO2 pressure which will be generated, so be generous with the caulk there).
Run the CO2 hose into the aquarium, through a one-way check valve, and terminating in an airstone in the aquarium. Leaving an air hose unattached at both ends, with one end in the aquarium, is an invitation to disaster. It is very easy for siphoning to start by accident. Capillary action draws aquarium water up the hose to the top of the tank, then an accidental yank on the hose can easily pull the hose out enough so that this water in the hose fall below the tank water level and thus starting a siphoning action. I had 20 gallons of water on my living room floor in a few hours, started in exactly this way. Therefore I strongly recommend having a one-way check valve in the CO2 path. Note that most cheap air-hose check valves do not last very long in a CO2 line. Carbonic acid formed by the interaction of CO2 with water tends to dissolve the rubber membrane. You should spend a few more dollars and buy a check valve made for CO2. By the way, NEVER put a shutoff valve (even under control of a solenoid or timer) in the CO2 line of a yeast-generated CO2 setup. If the line is shut, the pressure will keep building until the bottle bursts--very messy.
For the reactor, use a large-diameter plastic bottle (such as a Gatorade bottle). Cut out 3 large panels from the lower portion of the bottle. Insert the CO2 hose through a hole in one of the panes between 2 panels (see picture), near the bottom, and cap it with an airstone. Use stones to weigh the bottle down and place it in a rear corner of your aquarium. Place the water filter return spray bar vertically next to the bottle (secure to side of tank with suction cups). The idea is for the CO2 bubbles to come out of the airstone and collect at the top of the reactor bottle. The water from the spray bar then constantly agitates the CO2 surface and helps dissolve the CO2. The cap of the bottle allows
it to be bled occasionally, to remove accumulated, undissolved gasses.
Now mix the yeast, sugar and water solution as follows.
Put 1/4 to 1/2 teaspoon of yeast into the bottle with about a cup of warm water (bread yeast is fine). Some people have theorized that champaign or wine yeast should last longer (due to its ability to tolerate the alcohol generated from the brewing process), but recent tests reported on the net have not indicated any difference.
Add water to bring the solution up to 3/4 of the bottle.
Add 1 to 2 cups of sugar and shake well. The amount of yeast and sugar will determine the rate and duration of CO2 generation. More yeast will result in stronger CO2 production, but will exhaust the sugar quicker. Using 1/4 teaspoon of yeast and 2 cups of sugar will result in CO2 production for about 4 to 5 weeks.
In areas with soft water, some people recommend adding a teaspoon of baking soda to buffer the water and extend the life of the solution (prevent the acid formed by the brewing action from destroying the yeast prematurely).
The brew should generate about 1 bubble per second (from the air hose with no airstone), after about one day. Using warm water will help it get going faster. There is a ramping up period in the flow at the beginning and a ramping down period at the end of the solution's productive life. To smooth out the flow, you can use two smaller (1 liter) bottles, instead of one 2-liter bottle, and start one about a week or two after the other (but this doubles the maintenance effort). You should remove and clean the airstone occasionally, as yeast-generated CO2 has a tendency to generate a slime coating that gums up the airstone after a while.
If instead of using the bell-type reactor described here, you choose to feed the CO2 line into a powerhead, as some people do, then you should consider using a more rigid bottle (e.g., juice bottles) than the standard softdrink bottles. The idea is to prevent to bottle from collapsing if there is a suction on the CO2 line (by the powerhead). This could push the yeast solution into your aquarium.
Occasionally you will read on the web or in the newsgroups that you shouldn't shake the bottle to mix the yeast with the water. This is a classic net.non-sense which started with a post in the rec.aquaria newsgroup a few years ago. One guy posted the instructions for a yeast-CO2 set up, and wrote, "Add the yeast to the water, but do not stir or shake the bottle. The instructions from my bread machine explicitly stated this." What he failed to understand was WHY his bread machine instruction was so. Anyone with a bread making machine could see this specific instruction, under the "Delayed Baking" mode. That's the mode where you put the ingredients into the machine, then set the timer and go to bed. Near morning, the machine will automatically turn on and start mixing, kneading and baking, and you will have fresh, hot bread as you get up in the morning. The instruction tells you to put in the water first, then gently put flour on top of the water, then put the yeast on the flour, an DO NOT MIX. That's because they don't want the yeast to get wet and get activated too early, resulting in bread that rises too much.
Anyway, this guy did not understand the reason for that instruction, but passed it on anyway. People read this guy's tip and started passing that piece of non-sense around. It made its way into archives, and occasionally pops up here and there. I always have a good laugh when I see it.
Since we want our sugar solution to start fermenting as soon as possible, please by all means, mix the yeast well. But don't shake the bottle after the cap with the CO2 hose has been screwed on. This might get some of the liquid into the line, which will be pushed into your aquarium once the CO2 starts producing.