Current is the rate of flow of charge per unit time due to an [electric] potential difference.
What do you mean by what makes the current? Do you mean what is electric current? It's the movement of electrons in the conductor. A quick look on Wiki tells you that. First line. To be fair, it does seem that you're asking a lot of questions which could be easily answered by a quick visit to Wiki, or looking it up in a book.
As an afterthought, perhaps you meant why do the electrons move? It's because of the electric potential difference set up by the battery/cell/whatever, and so the electrons are attracted to the positive side of the battery.
Caution, though, conventional current flow is defined in the opposite direction as the flow of the electrons.
And why there can't be electric current created without closed circular loop?
Because then there would just be a build up on the end of the wire. Having a closed loop means the current can flow continuously. The electrons are attracted to the positive side of the battery, and then, once they have gone through the battery, the electrons are then push (similar charges repel) the electrons back round the closed loop again. (This is probably a very simplistic view, and it'll get picked to pieces, I'm sure; but it works for me to help visualise it)
Remember, current is the rate of flow of charge per unit time, so for a current to be maintained, the electrons need to be constantly moving - which can only be achieved by a closed loop (since there are only a finite amount of electrons in the wire).
And what is the roll of the magnetic and electric field?
do you mean "curl" or do you mean "role"? or something completely different?
and are the electrons jumping from atom to atom in coductor with electric current inside?
Please read up on valence electrons and shells:
But thats a rough idea, yes.
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