Тут is an adverb of location "here".
Я тут = I am here
Вот is a demonstrative, translated as "here" but not referring to the location of something, but rather pointing to its existence or being of the noun referred to.
Где машина? = where is the car?
Вот здесь = it is here
Вот там = it is there.
Здесь is also an adverb of location (you will have to look further for the explanation as to what is the difference between здесь and тут, I have not gotten a good explanation as to when to use one or the other, so far, for me is instinct). It shows where something or someone is.
Я здесь, в Москве = I am here, in Moscow.
Сюда is an adverb of motion. It does not indicate where something or someone is, but where something or someone is going.
Пожалуйста, идите сюда = please, come here.
You will find that just as здесь and сюда are pairs, там (location) and туда (motion) are also pairs.
Ты там, в Москве? = Are you there, in Moscow?
Пожалуйста, идите туда = please, go there.
[adsense:block:adsense-under-respons]