What does Judaism have to say about dreams and do have any legitimacy?
Are they in any way binding, or are they assumed to be just products of an overactive imagination?
Dreams are essentially a lower-prophecy that anyone can receive. In the Torah, many people experience dreams with messages, Yaakov, Yosef, Pharoh and his wine-maker and baker, Avimelech, Ballaam.
Talmud Bruchos 55a-57b has a bunch to say about dreams and how to interpret certain symbols in dreams.
on Interpreting Dreams: A dream is not inherently good or bad. "All dreams follow the mouth"(55b), if the interpreter interprets the dream as good, it will be good, if they interpret it as bad it's bad. As such, one should always try to interpret a dream as good. Bar Hedya would give positive interpretations only to those who paid him, and unfavorable interpretations to those who didn't. (56a)
Birkat Kohanim has a paragraph to pray for a positive-conversion of a dream you dreamt and you don't remember. (55b)
Rambam: in the Guide of the Perplexed
The other answers were very good, but if you want a lot of information, I recommend the book Mishnat HaChalomot by Rabbo Boaz Shalom. Rab Yehuda Fetaya (ZSWQ"L) (60-70 years ago) wrote that dreams don't mean anything any more, only under very specific circumstances. The Sefer Hassidim (700 years ago) also write this. I heard HaRab Boaz Shalom Shlit"a say in his speech that even if the dream meant something, there really is nobody in our time who can interpret it. In conclusion, quoting one of the Rabbanim of Lublin, he said that dreams are the inner deepest desires of the heart coming into fruition in a dream. Thus, dreams should not be taken seriously as much as they used in the time of the Gemara. I really recommend this book, it is very insightful with 750 pages and discussions on demons, dreams, ayin hara, and all that stuff.
It is a dispute, and different rabbis had different things to say.
Talmud Berachot 55b records an opinion that one's dreams are merely a reflection of what one is thinking during the day. Thus:
And on the next amud:
Such dreams would not seem to be prophetic.
Abarbanel discusses how some dreams are purely the result of the imaginative faculties, modified by the body's physical state, based on the four humors. Thus:
Of course, Abarbanel held that certain dreams were prophetic in nature. And of course, there are other views encoded in the Talmud, in close proximity, as reflected in other answers on this page.
But I think the most comprehensive answer would be that it is a machlokes; that there were rationalists and non-rationalists even during the time of the Amoraim; that in the time of the Talmud, dream interpretation was regarded as a science, and that this might change as our understanding of science changes.
So, to address the question as posed,
I think the answer is "yes".
We believe in dreams, as we see by Paaroh, and as we see there is such a thing as a Taanus Chalom - which is a fast when someone has a bad dream. At the same time the Gemara in Berachos 55:2 says that most dreams are from the thoughts we have. The Posuk in Zecharya 10:2 says וַחֲלֹמוֹת הַשָּׁוְא יְדַבֵּרוּ which means they have no meaning. I do not remember who says this, however there seems to be a difference whether an Angel or a demon showed us the dream. How to know the difference I do not know.