Both of these characters first appear in the writings of Geoffrey of Monmouth. Merlin in The History of the Kings of Britain (ca.1135-1138), Morgan in the Vita Merlini (ca.1150). Geoffrey says that his The History of the Kings of Britain is a translation of a "certain very ancient book in the British language." This is something of a trope in the middle ages and it is relatively unlikely that such a book existed. It seems instead that it is a composite of a number of different texts, of varying competency and historical value. (There's a lot of interesting things that could be said about exactly where these stories come from, but I'll try to get to your questions rather than going off on a tangent.)

In The History of the Kings of Britain, King Vortigern is having some trouble building fortifications and his magicians tell him to find and sacrifice a boy with no father. Merlin is found and identified as fatherless (since his father had been a demon). However, he turns the tables on the magicians, reveals the real reason the walls keep falling and gives a whole series of prophecies (all of Book VII and scattered throughout the following books as he continues to advice kings). Prophecy is Merlin's big magic throughout the Arthurian tradition, but he can also do a couple of other things. He helps Uther seduce Igerna through concoctions that alter the king's appearance to look like her husband Gorlois. He specifically states that these arts are "new" and "unheard of," which are caveats that probably mean such magic was not taken seriously by its audience.

Morgan le Fay is really interesting. To keep it simple though, basically the tradition takes her from being the healer who tends Arthur after his final battle in the Vita Merlini to being a totally evil witch in Le Morte d'Arthur where she is associated with death, illicit sex, wrongful imprisonment. Partly this is a result of the character moving from the more gender neutral (perhaps even more feminine) world of folktales to male courtly texts. Certainly, there were women healers in Welsh villages of this time, but the powers that she acquires in later sources are much more fanciful and associated with Christian demons.

Another interesting note is that the character Morgan in the Vita Merlini is the same character as the lady of the lake and this is likely the origin of her "fay" moniker. Later tradition splits the character.

These are really super interesting characters, but I feel like this has already turned into a wall of text, let me know if there's anything you'd like clarified.

edit: So in response to some comments, I will expand a little on the textual question (basing my answer largely on information from Michael Faletra's translation of The History of the Kings of Britain which has a brilliant introduction and appendices if you want to learn more).

Geoffrey of Monmouth was the son of a Breton settler born a generation after the Norman Conquest. He lived near the Welsh territories which the Normans were trying to annex which was the stronghold for the remaining Welsh/Cornish Britons who had been driven west by the Anglo-Saxon invasions some centuries before. Breton and Welsh languages were similar enough that Geoffrey likely could have understood some of the folktales from the region where he was raised. Geoffrey claimed that his work was a translation of a book given to him by Walter of Oxford, but comparison to contemporaneous Welsh works (like the wonderfully bizarre Mabinogion) makes this unlikely to be the case. Textual comparison reveals that Geoffrey was instead drawing on the Anglo-Latin work of Gildas (6th cent), Bede (8th cent), and Pseudo-Nennius (9th cent). Still, Geoffrey creates vivid and strong leaders for the Britons that were so popular his work was eventually translated into Welsh and became the basis for Welsh bruts.