Terraformers increase the agricultural potential of a planet. Your empire's total agriculture determines how fast your planet populations will grow, and even whether you will be able to attain your maximum possible population.
Terraformers add ag resources to the planet equal to their current BR multiplied by 10. Thus a full strength BR-3 terraformer will add 30 ag points while a BR-3 ship brought down to BR-1.8 will only add 18 ag points. Note though, that a planet's total ag resources can only be increased to the maximum value of its minerals or fuel, whichever is higher.
The terraformer will be used up in improving the planet, so you may need several ships to terraform a given system up to its full potential. Note, though, that you cannot terraform the same system with more than one ship in the same turn. If you set two or more ships to terraform the same planet only the one first on the ships list will actually work, the others will revert to standby mode. This means, then, that you should produce terraformers in a string rather than in groups so that they arrive one after the other.
No terraforming should be done at least until you reach BR-2, and preferably BR-3. To terraform with a BR-1 ship does no real good and is a waste of resources. I personally tend to start terraforming at BR-3 (or higher) unless the map dictates that I need to start at BR-2.
Terraformers have gone in and out of favor, depending on the dominant strategies among players at the time. In the past the game was much slower paced and more conservative, which led to more terraforming. The current trend is toward quick overbuilds and aggressive attacking, which makes terraforming difficult or even unnecessary. In these types of games you don't want to waste resources that could cost you one more ship at the front, and defenders are so hard pressed to stop the attacks that they often can't afford the resources to build terraformers. Most veterans will still try to get some terraforming done, but it is often in the mid-game rather than close to the beginning. The way I usually play, the trade bonus from 1-2 allies plus what you get from nuking and colonize enemy planets is usually enough to last until the mid-game or even all the way to the end. Terraformers tend to be my 4th-5th tech choice unless other circumstances arise.
There are two common and important misconceptions about terraformers, both of which I have noticed even in veteran players. The first has to do with the economy boost they generate. Adding ag does indeed increase your economy as seen on the diplomacy screen, but this is actually a misleading increase. The only things that matter to your ability to build and maintain ships is minerals and fuel. Agriculture only helps your planets to grow faster. If your planets are already at maximum population, or if you have enough ag to get there and are satisfied with your population growth rate, there is no reason to terraform further. The addition you see added to your economy score doesn't help build ships or fight, and the use of valuable resources to build more terraformers can actually hurt you in the long run. (Note, however, that the visible economy score very well may be an important psychological weapon so don't totally discount it.)
The second misconception is that ag is like minerals and fuel in that you only produce as much as you have population. This is true of minerals and fuel, you do only produce as much on a given planet as you have population on that planet. This is not true of agriculture though —so long as you have any population on a planet its full agriculture value is avaliable to you. You therefore only have to set your planet's population to the higher of the mineral or fuel values (or 50 if you want to make the planet a builder and neither value is that high) and NOT the ag value. Setting the population higher will only slow down the population growth of other planets.