One of the common misconceptions about Celtic music (not to mention all sorts of other genres) is that it contains different chords and/or scales than other music. Now while Celtic music may have a lot of “character traits” for lack of a better term, it still uses the chords and scales with which you are already familiar.
Most traditional music, including Celtic, was old long before the guitar even came around. Hence a lot of the flavor of Celtic guitar style comes from trying to get your guitar to mimic the nuances of the older instruments associated with the Celtic traditions - which would mean a lot of harps, fiddles and pipes. Having a lot of open, ringing strings creates both the sound of the harp and the drone of the pipes. This is why a lot of guitarists who specialize in Celtic music prefer either drop D or DADGAD tuning (which you can find out about in the column On The Tuning Awry), it gives a lot of opportunities for open drones in fifths. Say you're playing a solo in the key of D. You can use your lower three strings as a drone and play a melodic lead on your high strings at the same time.
A further way to imitate the pipes is to use a lot of “trilling” effects - such as hammer-ons and pull-offs. If you listen to Celtic music (and listening to any traditional music is the best way to come up with ideas) there is a lilting quality to it. Melodies flit about in a very ornate style. You cannot add too many trills. Another cool technique is to use vibrato on various notes. Not typical guitar vibrato (where you slide your finger back and forth along the string) but rather hard vibrato where you move your finger PERPENDICULAR to the string. This will cause your melodies to occasionally go slightly sharp or flat, but this again gives the impression of the instruments not being perfect. Well placed bends can also do this.
On Guitar Noise
Since September 2005 Doug Sparling has favored Guitar Noise with the following series of Celtic Music for Guitar lessons.