This is a surprisingly interesting question.
How much could have been in the movies?
Gandalf's origin is hinted at in the Appendices:
When maybe a thousand years had passed, and the first shadow had fallen on Greenwood the Great, the Istari or Wizards appeared in Middle-Earth. It was afterwards said that they came out of the Far West and were messengers sent to contest the power of Sauron, and to unite all those who had the will to resist him; but they were forbidden to match his power with power, or to seek to dominate Elves or Men by force and fear.
They came therefore in the shape of Men, though they were never young and aged only slowly, and they had many powers of mind and hand.
The Return of the King Appendix B: "The Tale of Years" (ii) The Third Age
Obviously this is still pretty vague, but it tells us two important things1:
- The Istari came out of the West, which is basically Elvish Heaven
- They're not Elves, Men, or Dwarves (or anything else we've seen in the books), but something else entirely; something that can change form
There are a few other winks and nudges throughout the series, like Gandalf's line at the Balrog:
'You cannot pass,' [Gandalf] said. The orcs stood still, and a dead silence fell. 'I am a servant of the Secret Fire, wielder of the flame of Anor. You cannot pass. The dark fire will not avail you, flame of Udûn. Go back to the Shadow! You cannot pass.'2
Fellowship of the Ring Book II Chapter 5: "The Bridge of Khazad-dûm"
But these are clues that can't be deciphered within these four texts. There simply isn't enough available information for the movies to draw any firm conclusions about Gandalf's true nature.
Why didn't this make it into the movies?
Well some of it did; see Gandalf's confrontation with the Balrog, which survived mostly intact in the film. As has been discussed a lot on this site (here, for example), the Tolkien Estate only sold off the film rights to The Lord of the Rings and its Appendices, and The Hobbit; any material from The Silmarillion or Unfinished Tales would have resulted in a swift lawsuit.
The fact that Jackson had the rights to the Appendices is pretty important; it's what let include the Sauron subplot in The Hobbit trilogy, for instance3.
However, it seems to have been a deliberate move of Tolkien's to exclude the true nature of Wizards from the text. In a Letter 153, when discussing Treebeard, he writes:
[Treebeard] does not know what 'wizards' are, or whence they came (though I do, even if exercising my subcreator's right I have thought it best in this Tale to leave the question a 'mystery', not without pointers to the solution).
The Letters of J.R.R. Tolkien 153: To Peter Hastings (Draft). September 1954
It's hard to argue that this was a bad decision, narratively. Trying to explain what Gandalf actually is would introduce a lot of unnecessary complexity to the book, and with The Lord of the Rings Tolkien really was trying to write a story, not a mythology.
It makes sense to exclude from the movies for a similar reason; your average movie-goer isn't going to know who Eru Ilúvatar is, or what a Maia is, and you shouldn't expect them to.
As Abulafia points out in a comment on the question, there's at least one hint that Gandalf is more than he seems. In The Two Towers, when he leaves Edoras to find Éomer, he remarks:
Gandalf: Three hundred lives of men I've walked this earth, and now I have no time.
The Two Towers (2002)
While this line doesn't unambiguously identify him as a supernatural being, it does at least suggest that he's not human.
At least one hint at Gandalf's otherworldly nature made it into the movie, when he discusses with Pippin about death4:
Pippin: I didn't think it would end this way.
Gandalf: End? No, the journey doesn't end here. Death is just another path. One that we all must take. The grey rain-curtain of this world rolls back, and all turns to silver glass... then you see it!
Pippin: What? Gandalf? See what?
Gandalf: White shores. And beyond: a far green country, under a swift sunrise.
Return of the King (2003)
The way Ian McKellen plays this scene suggests to the audience that Gandalf remembers this place he's describing, which is evidently Aman. This is more of a hint for fans of the books; as jpmc26 points out in comments, if you're not familiar with Gandalf's nature this just sounds like a memory of his own recent death.
1 I'm deliberately oversimplifying these points; the greater Tolkien canon tells us quite a bit about the nature of Wizards, but this information is stricly including sources that would have been available to Jackson and Co. i.e., what we can determine from the Lord of the Rings and Hobbit books.
2 If you're curious about what this means, there are some excellent answers here
3 Determining whether this was a good thing or not is left as an exercise to the reader.
4 As Eric Lippert points out in comments, Gandalf's lines here are lifted from narration in the book; this description appears twice in Lord of the Rings, first when Frodo has a dream in Tom Bombadil's House:
That night they heard no noises. But either in his dreams or out of them, he could not tell which, Frodo heard a sweet singing running in his mind; a song that seemed to come like a pale light behind a grey rain-curtain, and growing stronger to turn the veil all to glass and silver, until at last it was rolled back, and a far green country opened before him under a swift sunrise.
Fellowship of the Ring Book I Chapter 8: "Fog on the Barrow-Downs"
And again at the very end of the trilogy, when Frodo leaves Middle-Earth:
And the ship went out into the High Sea and passed on into the West, until at last on a night of rain Frodo smelled a sweet fragrance on the air and heard the sound of singing that came over the water. And then it seemed to him that as in his dream in the house of Bombadil, the grey rain-curtain turned all to silver glass and was rolled back, and he beheld white shores and beyond them a far green country under a swift sunrise.
Return of the King Book II Chapter 9: "The Grey Havens"