Please be advised that this is an over simplistic view.
There are many different models of hypnosis, some distinguish between hypnosis and trance. Some claim you are always in a trance in one form or another, and some claim there is no such thing. I use the word models because they are just that. Hypnosis and trance are not physical things. Yes you can see or hear the effects but as others have pointed out these effects can be attributed to other phenomenon. If you do relaxation hypnotherapy an onlooker could say the person is asleep, in the case of stage and street hypnosis people often ask are they acting. So people build models to try and explain what is happening and use words like the unconscious or subconscious mind, the higher self. A model is used to represent what is happening.
What Is The Unconscious Mind?
Yep here comes another model. As baby you have no conscious mind, you cannot control your hands or your legs but you do have some automatic systems running. These we will call programs, a term from NLP. You know how to breathe, your heart keeps on beating and various other things but even these programs may not be fully developed yet. As you grow you learn to take control of your hands and legs and at some point they become automatic. Now if you wanted to walk or pick a glass up you don't do it consciously except where there may be physical or neurological damage. This automatic response is what is deemed the unconscious mind. It goes much further than that but that's a basic idea. Emotions, memories, beliefs and habits all fall under the unconscious mind. So your unconscious is in charge of running all of your automated programs, even the ones that you learned.
What Is The Conscious Mind?
We develop a conscious mind to help us through life. Think about children, I taught one of my nephews how to say yes when shaking his head and no when nodding. It was amazing how fast he picked this up considering everyone was showing him the correct way. It was a bit of fun and he got reactions from everyone. So the conscious mind starts to develop to filter incongruent information. Go to an adult shake your head and say this is yes, they will most likely disagree. So information is filtered through our conscious mind to see if it we choose to believe it.
Does It Matter?
Actually no, people have be Hypnotised and hypnotic phenomenon gained (hands stuck together, stiff arm etc.) and still don't believe they have been hypnotised. They may say things like I could of opened my hands or bent my arm I did not really try. As a hypnotherapist all I ask of my clients is that they follow my directions. I have had people say it's not real but then you ask them to look at their hand, they keep looking around and you know they are trying to prove you wrong. Now there are ways of getting around this but I am here to help people who want to be helped and as I said before even if you do they will just deny it anyway. One of the fastest changes was with a client who did not ever feel like they were hypnotised.
What Is Hypnosis?
Hypnosis is a specific form of communication. A communication that bypasses the conscious mind, it is something that we are all capable of. Ever had a conversation with someone and at the end you realised that you had not been aware of your surroundings, ever drove or walked somewhere and cannot remember how you got there. These are all times when your conscious mind have been either bypassed or not needed, as hypnotists we communicate with that part of your mind that runs your automatic programs.