Unlike some other story tellers, Tolkien lets his magicians be subtle in wielding their magic. When they do great deeds with magic, it is mostly in stories of old ages, which are told within the stories we read. Rarely a powerful being directly performs impressive magic. This is true for Galadriel as well. She is powerful (allegedly she destroys the walls of Sauron's stronghold Dol Guldur all by herself), but what we are shown directly in TLotR and TH is her scrying, foretelling, and handing out subtly magical items.
Galadriel is a grandchild of the two kings of two of the great elvish dynasties (Finwë and Olwë), and closely related to the king of a third (Ingwë), and said to have inherited positive traits from all of her ancestors, making her the fairest and mightiest of all elves on Middle Earth (after Gil-galad's death).
She is among the Noldor when they rebel against the Valar and leave Valinor, and is said to be the only one who — at the end of the Third Age — had her ban lifted and allowed to return. At that time, Galadriel is well over 7000 years old and has been on Middle Earth since the First Age, for thousands of years before Gandalf arrived there. Galadriel sends Gwaihir, lord of the eagles, to search for Gandalf after he defeated the Balrog, she healed him, and clothed him in white robe.
She has always been admired both for her beauty and her ability to look into others' hearts and minds, and judge them (fairly, it is said) and was able to communicate telepathically. (She does so with Elrond, who is her son in law.) Due to her abilities, she (together with Gil-galad) distrusted the loremaster Annatar, who taught the elves to forge the rings of power, and was later unveiled as Sauron in disguise. Celebrimbor, who forged the rings, gave her the ring called Nenya.
While the One Ring was lost (and thus Sauron could not use it against the other ring-bearers), Galadriel used Nenya to turn Lothlórien, where she and her husband Celeborn ruled, into a safe refuge for the Silvan Elves. Later it became somewhat of a center of the opposition against Sauron's schemes. Allegedly due to the ring, Lothlórien withstood several sieges by Sauron's armies, Galadriel later played an important part in the destruction of Dol Guldur, from where those armies came.
When Frodo comes to Lothlórien and offers her the one ring, she is able to resist which is no small feat for a powerful being as she is. This is what makes the Valar lift the ban that prevented her from returning to Valinor. Thus, she departs to the West after several millenniums.