The purpose of the truss rod is to counter the pull of the strings and to maintain relief in the neck. Relief is the slight forward bow that allows notes to be played without buzzing. Truss rod adjustment is not hard to do, but must be done carefully. The results of over-tightening a truss rod can be disastrous, including a broken neck, broken fingerboard, broken truss rod, etc.
The first thing to do is to make sure that you have the right tool. Most acoustic guitars are adjusted with an allen wrench. Make sure you have one that exactly fits. (They often come with the guitar) The nuts on electric guitars can be much more varied. If this is an adjustment that you will be doing on different guitars, Stewart MacDonald's has a reasonably priced truss rod wrench assortment. They also have individual wrenches for some of the major brands.
The next thing to do is to determine if it needs be tightened or loosened. If you are unsure, see the page on setup. (There is a description there on how to check the neck relief.) A quarter turn is often all that is needed. If you need to tighten the truss rod, loosen the strings first. This will help prevent damage to the threads and allow the adjustment to set into the neck. After tightening it a quarter turn, tune back to pitch and check the relief again. If the nut will not tighten, try loosening it first. If it loosens, but will not tighten beyond where it was originally, it may be at the end of the threads. If this is the case, then the truss rod will need to be repaired or the neck heat-straightened. Never force a nut to tighten. If you hear any cracking or popping sounds, stop! Take your instrument to an experienced repair person and bring your checkbook. If the nut was snug to begin with, you shouldn't have to tighten it over 3/4 turn.
If the truss rod needs to be loosened, the strings can remain at full tension. This will help the adjustment to set into the neck. Try a quarter turn and then check the relief. Sometimes it helps to gently lift the head of the neck while holding it down in the middle. This should only be done with a few pounds of pressure, too much could damage your guitar. This should bring results quickly. If there is no change, then try another quarter turn. If this does not bring a change, the neck may need to be heat-straightened.
If a truss rod adjustment does not change the relief in the neck, heat straightening can usually fix it. This involves clamping the neck to a special tool that heats the neck. Besides actually bending the wood, the procedure also allows the glue under the fingerboard to soften and shift slightly. The neck is then allowed to cool while still clamped to the tool. This will cause it to set into a new shape with either more or less relief depending on how the tool was setup. This can cause the truss rod to be more responsive.