What is the meaning of the prefixal 'z'?

I mean in words like robić and zrobić, or jeść and zjeść What does that z do?

I know that in certain languages "to do" and "to make" are the same word (e.g. Italian fare). But what is that z in Polish?

Some examples:

będę robić - I will make
pójdę zrobić - I will go and make

będę jeść śniadanie - I will eat breakfast
pójdę zjeść śniadanie - I will go and eat breakfast

What is będę and pójdę? (reflective action and future action)?

I still don't really understand.

z = with

będę spał - I will be sleeping
pójdę spać - I will go to sleep

Robić is the standard dictionary form being imperfective and zrobić is the same thing but is the perfective form of the same verb. You must remember not to use the form zrobić meaning to do or to make in the present tense. Therefore, ja robię-I am doing or making and ja zrobię meaning I will do something or I will make something. In the past tense, ja zrobiłlem coś means I did something and the action has already taken place. On the other hand, ja robiłlem coś means that you were doing something in a general sense and that action has still not been completed.

Będę is I will from the verb być
Pojdę is I will go from the verb pojść.

in these examples, like Michal said, "z" changes the imperfective verb into a perfective one, the subject of perfective/imperfective has been discussed quite a lot on these forums in the last months, maybe you should look for some older threads (although it may be too early for you to bother with this, so bookmark them for a future reference).

Thanks so much for your help.