Galadriel is one of the last of an ancient people called the Noldor still living in Middle Earth. The history of the Noldor is the subject of the majority of the Silmarillion and, while Galadriel comes into it only briefly, the achievements of the Noldor can give us some insight into Galadriel's influence and power.
I've noticed that in the following, I talk about this nebulous term "power" quite often; let's spend a little time defining that.
In Tolkien's world, there are two 'layers' of reality - the primary world, and the secondary world. The secondary world is what Aragorn (in the films) meant when he said
"He's passing into the shadow world. He'll soon become a wraith, like them".
Elves, wraiths, and other supernatural creatures dwell largely in this secondary world, and this is where most of their power lies. It's the world that Frodo saw when he was dying from the stab wound from the Nazgul blade - that's why Arwen appeared to glow when he looked at her. In the books, it's described thus:
With his last failing senses Frodo heard cries, and it seemed to him that he saw, beyond the Riders that hesitated on the shore, a shining figure of white light; and behind it ran small shadowy forms waving flames, that flared red in the grey mist that was falling over the world.
The shining figure is the elf-lord Glorfindel; the shadowy ones are Aragorn and the hobbits. It seems that fire also exists strongly in the secondary world, which is why the Nazgul fear it.
This Power takes many forms, but in general it can be seen as the ability to shape the world to your will in some way or another. This may mean great skill at making things, or enormous charisma, or great physical strength; or, in some cases, all of them at once.
The Noldor were perhaps the most powerful people ever to walk Middle-earth. The Silmarillion, the great history that underlies the Lord of the Rings, is largely an account of their deeds and their history. To give you an idea of what they could accomplish:
The palantiri, the seeing-stones, were crafted by the Noldor; Gandalf speculated that they were made by Fëanor, the greatest craftsman who has ever lived. Fëanor was Galadriel's half-uncle (that is, her father was Fëanor's half-brother).
Thorin's sword Orcrist was forged by Noldor smiths.
Gandalf's sword Glamdring, which he used to fight the balrog and throughout LotR, was also made by Noldor smiths.
Frodo's sword Sting was a trifling example of the craft, but was made by the Noldor.
The Rings of Power, including the Nine Rings of the Nazgûl, the Seven rings of the Dwarves, and the Three of the elves, were all made by the Noldor, though with the help of Sauron.
Rivendell was among the last realms where Noldor could be found in significant numbers.
The Noldor are, in short, a Big Deal. Galadriel is accounted one of their leaders.
Living in Valinor
In brief, the first of the elves who awoke on Middle-earth were persuaded to travel into the West and live with the Valar - effectively the gods of Middle-Earth. Some elves grew tired or afraid on the journey west and remained in Middle Earth, but most made the journey all the way to the land of the Valar. They were divided into three groups or tribes - the Vanyar, the Noldor, and the Teleri.
In Valinor, the elves reached the pinnacle of their abilities. Galadriel was born during this time, the granddaughter of the King of the Noldor.
In time, Melkor - the first Dark Lord, and Sauron's old boss - planted suspicion in the hearts of the Noldor, making them think that they were being imprisoned in Valinor. Galadriel's half-uncle Fëanor eventually led a large portion of the Noldor to leave Valinor in pursuit of Melkor. Galadriel went with them, and we hear that:
Galadriel, the only woman of the Noldor to stand that day tall and valiant among the contending princes, was eager to be gone...the words of Feanor concerning Middle-earth had kindled in her heart, for she yearned to see the wide unguarded lands and to rule there a realm of her own will.
The First Age
At length the Noldor came to Middle-earth, drove back the servants of Melkor (now renamed Morgoth), and established their own realms. The Noldor were noted as being stronger, wiser, more skillful, and more powerful than the elves who had remained in Middle-earth.
This was the time that the power of the elves was at its height, the heroic age when legends were made. Galadriel remains the only woman mentioned as a leader of the elves in this time.
Near the end of the First Age, the Noldor kingdoms were overrun by Morgoth, and many of their leaders were killed.
The Second Age
At the end of the First Age, the western lands where the elves had settled were largely destroyed by the war between the Valar and Morgoth. Most of the leaders of the Noldor were already dead; many of the elves now chose to leave Middle-earth and return to Valinor. Galadriel, however, took a portion of the Noldor east, and with her husband Celeborn founded the elven kingdom of Eregion.
Eregion was the last great kingdom of the Noldor. Here the Rings of Power were forged, though Galadriel distrusted the being who called himself Annatar and taught them to make Rings. Her distrust was well-placed, since Annatar was Sauron.
Eventually, Galadriel left Eregion to take up the rule of Lothlorien; and there she remained, until the time of Lord of the Rings.
That, then, is the short version of Galadriel's history. Her great power and wisdom come from her history, being born in heaven and having lived through the greatest legends of Middle-earth. She's older than any other leader in Lord of the Rings or the Hobbit (except for Cirdan, the shipwright, but he never does much). She is in many ways a relic of a bygone age, a memory of what the elves could do at their peak.
Galadriel bears one of the Three, the pure rings that the Elves made without Sauron's personal help. The power of the Three is to preserve and remember, so it is likely that the ring helps her maintain her native power, keeping her at her peak.
As for why she was tired after she banished the Eye in the third Hobbit film, this is, of course, the invention of the filmmakers, but it's not entirely surprising.
Sauron is a Maia, one of the servants of the Valar. He is a being far beyond any elf or human, an angelic spirit that once helped to build Middle-earth in the very beginning. He has by this time lost much of his native power, pouring a lot of it into the ring, squandering more in creating servants, but he is still formidable. The Valar forbade any of their people from fighting Sauron directly, since last time they tried something like that it destroyed a continent.
All things considered, the banishment of Sauron from Dol Guldur was a great achievement for any individual (while it's true, as comments have pointed out, that Sauron planned to flee and return to Mordor, he also most likely could not have withstood Galadriel if he wanted to); Galadriel is one of the few who could possibly have driven him out.