If it is Sunday, we would say tomorrow. Any day before Sunday, we would say "on Monday" or "this Monday".

If it's the following day you would just say 'tomorrow'. Say we are on Monday and you are going to meet someone over a week later on Tuesday of the following week. You would say see you next Tuesday. If it's Wednesday and you plan to meet someone at the weekend, you'd say see you on saturday. If it's the start of the week and you want to make it clear you want to meet someone during the week then you would say see you this tuesday. We say 'the monday' quite rarely, you could use it in a question e.g. 'are you gonna go and see that film on the monday or the tuesday?' Or in reply to when are you doing something that is provisional and still being discussed, 'Oh I'll do it on the monday'. This whole area is pretty confusing, even for native speakers. Often you hear people say, do you mean this monday coming or do you mean the monday after next? Dates often need clarifying because there is some difference between different parts of the UK about the meaning of these terms. I live in the north of England by the way.