I don't really understand the difference between these stuff. They look really similar. What is the difference between those? Which one do people use in measure theory, and probability related things and why? Is there any advantage to use one over the other?
A $\sigma$-algebra of subsets of $X$ is a $\sigma$-ring of subsets of $X$ which contains $X$.
The axioms of a field of sets differs quite a lot from $\sigma$-things since it need not be closed under countable intersections or unions. But in any case, both $\sigma$-algebras and $\sigma$-rings are fields of sets.
$\sigma$ algebras are used in measure theory since they model the set of measurable subsets of a measure space.
Asking if one of these has an advantage over the others is a little strange, but I suppose you could say a field of sets has the advantage of being the most general, while the $\sigma$-algebra is the most specific (if you can call these advantages.) Each of them can be used in any situation where the axioms are suited.