First of all, make sure if your guitar needs adjustment or not. If the nut is too low, it will be self evident by the string buzz occurring when the strings are open (unfretted), but not on the fretted notes. If it is too high, the action will be stiff at the first position. Bear in mind there is no universally correct dimension for string height at the nut. Instead, it is necessary to set the string height at the nut based upon both theory and the particular preferences of the player.

Measuring Nut Action
Here you need a set of feeler gauges (the thin metal strips that are used for gauging the size of a gap. These can be obtained from any auto-supply  or hardware store.

Fret the string at the frst fret and slide a feeler guage under the string right in front (body-side) of the fret. If the feeler guage is loose, try progessively thicker guages, until a snug fit is found. Not too tight however - if the guage pushes up the string go down a guage thickness. You may need to use a combination of different guages to get the exact height.

When you have the correct guage, this should be a near perfect indication of how high the string is from the fretboart at the nut.

Lowering the Nut
If you found that the string height at the nut was too high, you need to cut the grooves deeper. The tools necessary for this are either a set of calibrated nut files or aset of needle files. Nut files are special files that are manufactured to cut a round-bottomed slot of a particular width.  While nut files are easy to work with and remove much of the guess work from nut slot filing, they are expensive, and not essential.

loosen the first string slightly (only the first - you need to keep as much of the normal tension on the neck as possible), so that you can lift the string out of its slot and slide it towards the second string, letting the first string rest on the top of the nut. Place the end of the gauge(s) on top of the fingerboard so that the edge of the gages are touching the nut. Just insert enough to reach to the second string.

Using either a nut file or a needle file, deepen the slot until you just begin to contact the top surface of the feeler gauge(s). The slot should be cut by filing  the slot at a downward angle from the fingerboard towards the head. Failure to cut at an angle can result in poor intonation and string buzz. The width of the slot should be slightly larger than the diameter of the string. Once you have filed to the correct depth, remove the guages, reposition the first string and tune the string.

Repeat this for the second and third strings.To adjust the height of the fourth through sixth string, insert the gages from the bass side of the fingerboard and repeat the same process.

Raising the String Height at the Nut
If you found that the string height at the nut was too low, or you cut the string slots too deep, the string height at the nut can be increased.

One common and easy way of increasing the string height at the nut is to place a shim of mahogany or maple veneer beneath the nut. To do this, remove the nut, cut a shim to the same width as the nut, glue the shim to the bottom of the nut and reglue the nut to the neck. Once the nut has been shimmed sufficiently, the string height can then be lowered, if necessary, using the same procedure as described above.