Magic has many forms in the Gargoyles Universe, but two specific variants stand out above others: mortal magic (that practiced by some humans and gargoyles), and the magic of Avalon (which is used by Oberon's Children). Greg Weisman has compared the two variants to playing chess or checkers: "The board looks the same, but it's a totally different game." 
Mortal magic is generally worked through spoken incantations. No language is any more magical than any other language; however, many spells are recited in an ancient language, such as Latin, because they were originally written in those languages.  A mortal magic-user's spell usually has to be activated through a magical object of some sort, serving as a conduit, such as the Grimorum Arcanorum.
Mortal spells often have counterspells, but these may be tricky to work out. In some cases it may be possible to reverse a spell by discovering a condition to break it within the very incantation, and fulfill that requirement through mundane means. Thus, Xanatos broke the Magus's curse of perpetual stone sleep upon the gargoyles by making Castle Wyvern rise above the clouds (i.e., moving it to the top of the Eyrie Building), and later, with the gargoyles' help, broke Demona's curse of "stone by night" upon New York City by making the skies burn (with the help of some jet packs filled with a harmless hypergolic gas). Elisa undid Demona's spell of mind control upon Goliath through a cunning stratagem; she ordered him to act for the rest of the life the way that he would if he was not under a spell. ("Awakening Part Two", "City of Stone Part Four", "Temptation")
According to Hudson, in order to be affected by magic, one has to both see and hear it; at least, this was the case with Demona's "stone by night" spell, which consequently did not affect Jeffrey Robbins because he was blind. ("City of Stone Part Two") It is uncertain if this is the case with all mortal magical spells, since the "spell of humility" that the Roman Magus cast upon all gargoyles certainly extended all over the world, far beyond the bounds of the Roman Empire, but may have been able to due so by combining some powerful artifacts.
Mortal magic isn't restricted to human use and can be practiced by gargoyles, although they do not take up the study very often (Demona and Una being currently the only exceptions known to us).
Actually making use of magic seems to be quite tricky, from the evidence; it is apparent that only a practicing sorcerer can handle a spell or magical device properly. (Hence Xanatos's refusal to make use of the Grimorum as a means of undoing Demona's "stone by night" curse.) Demona described the spell that she placed upon Goliath to control him as complex, even though it consisted of only a few Latin words. Presumably much of magic has to depend upon training in such matters as correct pronunciation and gestures.
Known practitioners of mortal magic include the Magus, the Archmage, Demona, Macbeth, Brother Valmont, Rabbi Loew, Una, the Aborigine Shaman who served as Dingo's mentor in Australia, the Mayan wizard who crafted the Mayan Sun Amulet, and the Roman Magus. Merlin may also have used mortal magic, although this is uncertain at present, owing to his halfling heritage.
Avalon magic is the inherent ability of all members of the Third Race. It is generally activated by speaking a rhyming couplet or two, although occasionally without words, or, in the case of Oberon, a single word. (During his "enhanced" period, the Archmage could also cast spells with but a single word, an indication of his great power.) Judging from Alexander's case, the same applies to halflings as well. ("Avalon Part Two", "Possession")
Avalon magic cannot directly affect anything made of iron. However, the Childen of Oberon can get around this restriction through such methods as striking iron objects with powerful gusts of wind (as Oberon did to Xanatos when the latter was wearing an iron exo-suit). ("The Gathering Part Two")
Further information about Avalon magic can be found in the entry on Oberon's Children.
Other Notes on Magic
According to Xanatos, "mixing magics is dangerous." Presumably, this applies in particular to mixing mortal magic with Avalon magic, although it could even apply to mixing two pieces of mortal magic, given that the context in which he uttered this statement was Owen's suggestion of using a spell in the Grimorum Arcanorum to undo Demona's "stone by night" curse upon New York. ("City of Stone Part Three") Greg Weisman has hinted that it may be a case of "Different frequencies" or "feedback" that makes mixing magics dangerous and has suggested that an explosion or "some other backfiring" would be the most likely result from doing so.  Thanks to the difficulty involved, a Child of Oberon is also not likely to interfere with a spell previously cast by another member of the Third Race. 
Beyond Avalon and mortal magic, there are other forms. The Space-Spawn may have practitioners of alien sorcery, although this has yet to be confirmed.  Ghosts, such as those of Hakon and the Captain of the Guard, have intrinsic magical abilities, such as being able to conjure up illusions or temporarily possess people. ("Shadows of the Past") Among the objects that can be charged with magic are: the Grimorum Arcanorum, the Eye of Odin, the Phoenix Gate, the Cauldron of Life, the Mayan Sun Amulet, the Praying Gargoyle, the Medici Tablet, the Scroll of Thoth, and maybe Excalibur. (Not every object in the possession of a wizard is necessarily magical, however, as witness the Scrolls of Merlin.) ("A Lighthouse in the Sea of Time") Certain places can also become charged with powerful magic, such as the Archmage's cave. Some such places act as a nexus of mystic energy and even gain a certain atmosphere of holiness, a sense of connection with God, and humans build temples and churches in such places.  As a result, some forms of magic need to be cast on holy ground, such as Demona's use of the Fulfillment Spell (which she cast in Saint Damien's Cathedral). ("Hunter's Moon Part Three") Indeed, there may exist a third, more spiritual form of "magic" that is related to the existence of a higher power. Angels and the Stone of Destiny (which would not consider itself a "magical item") may be related to this third form of magic. 
Further information about magic can be found in the specific entries on individual magic-users or artifacts.
Behind the Scenes
When making up the mortal magic spells used in the series, the incantations were originally written in English before they were translated (usually) into Latin. Greg Weisman notes that despite the crew's best efforts, there were "mistakes in the Latin for almost every spell ripped from the Grimorum". The Phoenix Gate incantation also suffered, with Deflagrate being more accurate than Deslagrate, the word which was ultimately used in show. On the other hand, the Hebrew spells featured in "Golem" are likely to be the most accurately translated, most significantly due to the fact that Hebrew, while ancient, is still spoken today.  This also explains why the spell from the Scroll of Thoth in "Grief" was translated from ancient Egyptian when as a language it has not been spoken out loud for more than 1,500 years.