While Rambam and Ramban agree on quite a lot of the fundamentals of religion and the way God works, sometimes, with some things — even some fundamental ones — Ramban (who lived after Rambam) disagrees quite passionately with Rambam's take on things. One of these disagreements is about the topic of angels.
(One note here: As you've probably realized, Rambam and Ramban have dangerously similar names, so read carefully.)
Parshas Vayeira begins with angels coming to Avraham. The posuk then says that God appeared to Avraham and that Avraham was sitting outside in the heat of the day when he saw three men coming towards him. There are order issues here, and God's appearance seems rather insignificant. It's stuck in the posuk there, without anything else really being said about it. So of course, we must ask: Why?
Ramban takes a crack at it. He starts, in fact, by quoting Rambam’s Guide in which Rambam explains that this parsha, or at least this incident with Avraham, is written in “klal u’prat” — or “moving from general, to more specific”. It says that God appeared to Avraham "generally", and in what “specific” way did he appear? In the way of three men. In this way, Rambam explains the entire episode as being merely a prophetic vision. Never were there any real people. Never was there any real interaction. It was all a vision, or dream of sorts, that only Avraham experienced. Rambam, in this instance, indeed seems not to take the Torah literally. (Of course, it is crucial to note that non-literal does not equal less important. Indeed, it could mean that it is more important as it is not a simple recounting of events, but rather created specifically to teach us something, but more on that another time.) In truth, this entire comment of Rambam meshes perfectly with his overall view of angels and how they operate — which is, quite honestly, very similar to Ramban's (again, more on that some other time) but here, specifically, Ramban is not at all pleased with what Rambam has to say.
And that is an understatement. In fact, Ramban is vehemently opposed to what Rambam says here about the angels. It simply makes no sense, Ramban posits. Why would the Torah say that God appeared to Avraham if it was men that did? By no other prophecy in all of Tanach does it say that “God appears” to the person! And, according to Rambam, Avraham never killed or prepared any cows, nor did Sarah ever laugh at the prospect of having a child. Indeed, the prophecy must extend rather far — all the way to story of Sedom, in fact, making a tremendous amount of the Parsha nothing more than a dream. It is just too hard to determine when the prophecy starts and stops, and there is just too much falsehood here if it was truly all just a prophecy. Furthermore, Rambam explains that Yaacov wrestling the angel was a prophecy as well, which makes it really hard to explain why the posuk then says that Yaacov experienced pain in his leg, if it was all just a vision.
Ramban finally hammers home his point with very, very strong words against Rambam in this particular instance. To paraphrase, Ramban says the following about Rambam's opinion on the matter:
These things simply contradict the text [of the Torah]. It is forbidden to listen to them, much less believe in them.
Of course, there is a rather valid defense of Rambam too. It is indeed possible that it was all a vision. Why can't it be? No, there were no real people. There were no real cows. There was no real laughing. Why is that so difficult to say, in the end? Indeed, even when it comes to the story of Yaacov and the angel, we do have an idea that people can create pain in their minds. There is a real psychological idea we have today that Yaacov could have woken up from a dream so real that he felt pain in his leg. Pain is, after all, all in the brain. Ritva says as much to defend Rambam's position here — 500 years before any real major study of this psychology, by the way.
Both Rambam and Ramban hold that angels, in general, have no form. They do not actually possess (only) one leg, nor do they have wings — as the popular mythologies go. What exactly angels are, then, is a discussion for another time. But needless to say, on this fundamental point, both parties agree. Where they disagree is as follows: Ramban says that anytime the posuk says that an angel is speaking, it is a vision. But whenever the posuk says that the angels are men, then it is an actual event that took place. Dabbling in Kabbalah, Ramban explains that an angel will, on rare occasion, "wear the cloak of a man". This is called a “malbush" — whatever any of that means. Point is that Ramban agrees with Rambam that angels do not generally take the form of a human — except for here (and one or two other unique places). It must have been some sort of exception.
There is no doubt, though, that discussion very much intersects with the more general topic of God’s interaction with Man; the interaction between the infinite and the finite.
The Or HaChaim comments about the strange wording of the posuk "Veyeira eilav HaShem" or "It was revealed in/to Avraham, God". It should be "Veyeira HaShem eilav" as "Veyeira" modifies God — "It was revealed, God, to Avraham". It shouldn't jump from God to Avraham back to God as the subject of the verbs. Oh, and there’s another problem here as well: what did God even tell him? There's a nice Midrash, of course, about Avraham being sick and God doing the mitzvah of bikur cholim here, but there isn't anything about that in the text. What, did God just pop in for a second and then leave for no reason?
It seems that this revelation was Avraham changing to be a vessel for God. He tapped into the part of God that resides in every human and thus God truly was revealed in Avraham. The Bris Milah that Avraham had was God's branding, of sorts, on the actual flesh of Avraham. Avraham himself was a revelation of God. (Indeed, the true definition of a Kiddush HaShem is not just making people think Jews are good people, but something much deeper than that. It should make people think that there might, in fact, be a God in this world. The real definition of a Kiddush HaShem is bringing God into this world.)
Later in the parsha it says that "the angels went to Sedom". Hold on a second. Why switch from calling them men to angels? Says Rashi: When God was with them, they were men, but when God left them, they were angels. But what does that mean? Or, Rashi suggests, only with Avraham, who was used to seeing such holy things, were they like men. But to everyone else, they were angels. But wouldn't you expect the opposite? Someone attuned to holiness would see them as angels, while everyone else would see them as just men…
This is all answered by what we are saying. For Lot and those in Sedom, seeing these beings was exceptional, and thus they were called angels. For Avraham, he knew that God Himself is revealed through humans. The closest we can come to God is ourselves and our tzelem elokim. It is easier to find God in us than trying to understand God in the heavens.
Ramban's idea of angels cloaked in man could be an introduction to this idea. Rav Shimshon Raphael Hirsch goes back to the question we posed earlier: When Avraham says "My Lord" or "Adonoi", was he talking to God, or the angels/men? He answers that, whatever the explanation, we can learn that God reveals Himself to Avraham as he's doing a mitzvah — Hachnasas Orchim. This is the nature of prophecy for a Jew. Most people assume that it is in shedding this world that prophecy is achieved, but that is not the case. It is in connecting our physical lives to God that prophecy and revelation is, and was for Avraham, achieved.
It is found in the depths of humans and in a full understanding and appreciation of creation.