I wold like to know what does it mean to say something is a set.

For me, any collection of elements is a set. Then this Russel paradox says collection of all sets is not a set. 

All this came when looking at locally small categories. They are the categories where Hom class of any two objects is a set. 

What is a set? What is a class? Do we loose anything if we assume all categories are locally small?