As an aside. All three are very similar words being applied to a very similar people; a branch of the big Celtic ethnoi. But all three have a completely different origin, which is a testament to the fact that just because words sound alike doesn't mean that they are.
The Gaels are the Irish and Scottish, who speak very closely related languages that (along with Manx) make up the Gaelic subfamily within Celtic. The word appears to have originally come from a very primitive proto-Welsh, though (part of a different subfamily of Celtic)—in Old Irish, Gael was Gaoidheal, from Primitive Welsh Guoidel which meant something akin to "wild men" or "forest people." Or maybe warrior/raider. Probably describes the relationship that they had with the Dalriadans.
Gallic and Gaulish, although sounding very similar and referring to the exact same group (the tribal confederation that occupied what is today France, and which fought a series of famous wars against Julius Caesar, during which Caesar came, saw and conquered, after all was said and done) are completely unrelated words.
Gallic comes to us from Latin Galli and Gal(at)ai (which as you can guess, also gives us the name for the Galatians, which were still a Celtic speaking population and ethnostate in Anatolia 350 years after its founding by wandering Celtic warriors. It probably is ultimately descended from a tribal name in use among the Celtic groups in question themselves, and Celtic etymologies have been determined and proposed for the name. Of course, the Celts were named Keltoi by the Greeks, and Julius Caesar said that the Gauls called themselves Celtae in their own tongue. Linguist Stefan Schumacher, on the other hand, suggests that Galli might be an exonym applied by the Romans that originally meant something along the lines of "raider, marauder, robber"—a landlocked equivalent to Viking, in other words. Given the Roman history with the Cisalpine Gauls long before the conquest of Gallia by Caesar, this might well have been accurate (c.f. Brennus of "Vae victis!" fame. No relation to the other Brennus who defeated a Greek force at Thermopylae, reputedly sacked Delphi, and when finally driven off, went into Anatolia to found the state of Galatia—or rather, the remnants of his army did. Brennus himself doesn't appear to have survived the attack on Greece. According to the Greeks, anyway.)
Gaulish, on the other hand, has a Germanic origin, and comes from a Frankish word walha which meant merely "foreigner." (In French, Frankish initial sound w is very regularly rendered with a g: Guillame = William, guerre = war, garder = ward, etc.) In Anglo-Saxon, walha was more like wælisċ which is where the word Welsh comes from. It also gives us the -wall in Cornwall—Cornovii (tribal name) + wall. The Cornovii-foreigners. It also gives us the names Wallace and Walsh, the ethnonym of the Vlachs and Walloons, Welschschweiz (the word in Swiss German for the French part of Switzerland) and even the word walnut (foreign nut; because it was grown by the Romans, not the Germanic peoples.)
Curiously, there is a theory that the origin of the word in Germanic itself comes from an early Celtic tribe, the Volcae (a branch of which the Greek raider Brennus represented). If so, it would have happened very early in the development of Germanic—before even the First Germanic Sound Shift, or Grimm's Law, took effect, because the effects of Grimm's law can be seen on the change from Volcae to *wolk- to *walh-.
As an aside, it is presumed that the Gallaeci who gave their name to the region of Gallicia in northern Spain and Portugal were a Hispano-Celtic speaking population, it is completely unknown what relation their name has to the the Galli or Galatai or anyone else.
The Celts really were quite integral to the formation of Europe as we know it in many ways, and yet in many ways, they are one of the great lost peoples of Europe; never literate and culturally subsumed by either (or both) Romans or Germanic peoples without ever having had the opportunity to give us their own account of themselves, except for the rump populations of Wales, Scotland, Ireland, and Brittany. It's curious, and I wonder how much the character of guys like the English or the Swiss are informed as much by the Germanic folk who gave them their language and name, and how much was informed by the Celtic substratum which gave them a fair bit of their genetics.