Does BB-8 in Star Wars: The Force Awakens speak some kind of binary (Morse-like) language? If so, how can people understand it (such as Rey)?
Do we know the structure of it?
In-universe it's referred to as "Binary".
The droid blurted out something harsh. Rey scoffed. Her comfort with all things mechanical had made her fluent in binary. “Classified? Really? Well, me too. Big secret. I’ll keep mine and you can keep yours.” - SW: The Force Awakens - Junior Novelisation
Q. Why is R2-D2 so hard to understand?
A. R2 speaks in binary, a language of beeps and whistles. C-3PO often has to translate to help everyone else understand what he's saying. - SW: Absolutely Everything You Need to Know
It's also referred to as "Droidspeak"
BB-8 converses in 27th generation droidspeak code, a compressed variant of the most common astromech language - Star Wars: The Force Awakens Visual Dictionary
You may wish to note that this form of communication appears to be a dumbed-down version that can be understood by humans. When BB-8 attempts to communicate with R2-D2, his greeting is described as having a...
...transmission sequence too rapid and too exhaustive for any human to follow.
Binary or droidspeak was the language commonly spoken by droids. It consisted of a set of beeps and whistles strung together into what resembled sentences.
The astromech BB-8 spoke in 27th generation droidspeak, a compressed variant of standard droidspeak.
Rey and few other also can understand them surely by learning. It must be like learning any other language.
In-universe they call it Binary, which one could perhaps compare to morse code. You can learn morse code. It's not so much a language unto itself, but a way to encode language.
Of course, in reality, the noises the droids make are clearly analog and not binary at all, so it's actually more like a natural language in the aspect. (Though, as far as I know, there is no actual language created for the film like, for instance, they did with Klingon for Star Trek).
Official DVD/Bluray Subtitles/CC simply call this Droid Language when it doesn't describe it as Beeps or Whistles. This is used consistently throughout Episode 1-3. Episode 4-6 does not contain much non-verbal droid communication aside from R2. Pictured, a Vulture Droid at the start of Episode 3: Revenge of the Sith.
The same Subtitles use the language name for any other language, including Huttese and Galactic Basic (or Just "Basic", aka English or whatever language the movie was dubbed into). So officially, it's Droid or Droid Language. Of course this will be confirmed on the Force Awakens home release.