What is the difference between cousins of first and second remote? Are there third remote and so on?

Yes there are first, second, third etc cousins, each number is one generation removed.

Your Aunt and Uncles child is your first cousin, your first cousins child is your second cousin etc.

In may places it is illegal to marry any blood relative closer then a third cousin because of the blood line. Children of that marriage may be born with mental disabilities due to the blood lines. Cousins further then third cousins are far enough removed that it's safer medically.

First cousins are children of your parent's brothers and sisters. Second cousins are children of your grandparent's children besides your parents. Third cousins would be their children.