First, let me dispel one myth, which I have personally seen destroy dozens of necks in the last few years: The truss rod is not for adjusting your action! True, it does affect the action of an instrument. This is a byproduct, rather than the reason for adjusting.

If I haven't scared you off at this point, good - you at least have the nerve to try and do things yourself, and with care, patience and the missive below, you have a great chance of success.

Tune your guitar to the exact tuning you use. Fail to do this, and you will get a faulty impression of the curvature of the neck.

Locate the truss rod nut. This will be at one end of the neck or the other (usually behind the nut, or at the base of the neck. Most often in the soundhole with acoustic guitars). If you can't find it, chances you have a classical guitar (which I should have mentioned earlier ;-). Remove the truss rod cover if there is one. 

Hold your guitar in the playing position. Push the  low E-string down on the first fret and the highest fret simultaneously and inspect the relief (gap) between the bottom of the string and the top of the seventh and eighth frets. There should be about half a millimeter. Not all guitars are equal and that some will need a little more or less than this amount.

Use the correct tool to do any required adjustment. This will be in the form of a spanner,  allen key, or in other cases, a screwdriver. If you don't have the correct tool, either get the correct one or stop right there. Use of the incorrect tool may ruin your truss rod nut. On this subject - many American made guitars will have Imperial size nuts (which in many cases are similar in size to their metric counterparts), these may be almost right, but do damage anyway.

If the relief is insufficient (less than 0.5mm clearance at the seventh and eighth frets, or none at all), loosen the truss rod a quarter turn in an anticlockwise direction.

If the relief is too large, gently turn the truss rod nut one quarter turn in a clockwise direction.

Do not make any adjustments more than a quarter turn at a time.

Check the relief at the seventh and eighth frets again, and if it is still not right, repeat the above process.

If the truss rod nut:

• Requires large amounts of adjustment...
• Makes "skroink" sounds which are more suited to Don Martin...
• Turns with no resistance...
• Turns with too much resistance...

...something is WRONG. Do not pass GO. Advance directly to your Luthier.

Once everything is right. Congratulations! Put the cover back and play. then check it again in 24 hours - the truss may have "settled" a bit and need an eighth of a turn or so to tweak it to perfection again.

Special Notes:

• Double truss rods - Don't even think about it, unless you have a degree in engineering and lots of practice on cheap, disposable guitars.
• Old or "reissue" Fender Guitars - with the truss rod nut at the base of the neck, obscured by the scratchplate. With these, you need to remove strings, remove the neck, adjust the truss rod nut, then put the neck back, replace the strings, retune, check relief, remove the strings, remove the neck, adjust the truss rod nut, etc. etc. etc....
• Left hand thread truss rods - these are rare, but obviously need a bit more attention and care to get right.