For instance the exclamation mark is used for emphasis but what is emphasis exactly?
I have read that it means special importance but what does that mean?
Emphasis is a general term used (as stated in a comment) to give special attention to a word, phrase or sentence. All natural-languages have mechanisms for emphasis. You should try to imagine how you would answer this question for your native language.
English can give emphasis by stress (saying a word or phrase louder or slower) intonation (the rise and fall of pitch), syntax (changing the order of words), word choice and a range of adjuncts that have an emphatic effect. (I’m probably missing some methods)
In text, emphasis can be made by syntax and word choice, but as there is no stress or pitch in written English, emphasis can also be indicted by font changes (bold and italic) and punctuation (notably the exclamation mark).
Whatever method is used, emphasis marks some parts of speech as being more important.
The exclamation mark in particular indicates that the sentence expresses a strong emotion (surprise, joy, anger etc) and would be spoken emotionally. There is no simple recipe here. I can’t say that it means the sentence should be louder, slower or intoned differently. You need to look at the context and infer the emotional state of the person.