Hypnosis is defined in lots of ways; people who do it and study it all disagree on what exactly it is, how it works, what it can do, and so on. Most people agree that it involves the subject being extra-bonus suggestible. What it theoretically can do is "anything that the mind can do." This isn't a terribly useful answer, though, because most people have absolutely no idea what the mind can do, and most people's minds can only do a subset of what the human mind in general is capable of.
Assume all the caveats about the subject consenting and the hypnotist crafting the appropriate suggestion and the two of them having a good rapport and all that jazz.
Hypnotic suggestions work very well for things that the person can do, on purpose, on their own. If you are capable of deciding to sashay across the room meowing like an Irish cat and then doing a somersault pratfall while singing a Russian drinking song about vodka, then a hypnotist can cause you to do that. If you don't know how an Irish cat meows or a Russian drinking song, you are likely to approximate it to the best of your ability... which can sometimes be pretty hilarious.
Hypnosis can, less reliably, affect things that a person's brain can do but that the person does not know how to do on their own. You know that your brain does lots of things automatically that you don't think about, like keep your heart beating. Things that happen in your brain affect your heart rate, and many people have a bit of skill at calming themselves down, but almost nobody can deliberately and arbitrarily alter their heart rate. I think of it as hidden buttons and levers in the mind.
Because the hypnotist is talking to your subconscious, and because your subconscious knows more about these hidden buttons and levers, a hypnotist can craft suggestions to do things that the subject's conscious mind cannot do. For example, my brain knows how to have an orgasm, but I can't decide "Ok, I will have an orgasm now!"* But Mr. Shiny can tell me to have an orgasm, and I do. His suggestion can flip that lever in my brain.
An ex of mine could change his resting metabolic rate, so that he produced more or less body heat. It wasn't just his subjective experience of heat or cold, a lot of people can do that. ("I'm cold!" "Think warm thoughts." It kind of works.) He actually changed how much heat his body produced. It was incredibly useful on cold winter nights. Hypnosis can do that, too.
Some people can have lucid dreaming, where they control their dreams. A hypnotist can change what you dream about.
There's a mind-body health connection, too. Conditions involving the immune system and conditions related to stress response can be affected by what the patient is thinking and feeling. Pain, in particular, is highly subjective and dependent upon the patient's mental condition. Hypnosis can affect all of these conditions.
All those wacky human tricks that someone in the world has figured out how to do using meditation or religion? Hypnosis can do that, too. But not for every subject, and certainly not for every hypnotist. Some people are more suggestible than others. People also have internal safeguards that prevent someone from mucking around, although there's techniques for getting around a lot of that.
Perhaps most importantly, what a hypnotist can do with a subject depends on what the subject believes, deep down at the bottom of their heart, that the hypnotist can do to them.
WARNING! Do not let an undertrained hypnotist attempt to modify your physical health or your pain response! If you are a hypnotist, do not do this without training. Mr. Shiny can make my asthma better... which means that if he screwed up one day, he could also make it worse.
* Well, I almost can. I'm learning a lot about coming by observing what happens when he triggers me.