Zermelo and Fraenkel expressed their axioms in the formal language of set theory, but we can translate them into ordinary language.
Axiom 1: Two sets are equal if and only if they have the same elements.
Axiom 2: There is a set with no elements called the empty set.
Axiom 3: For any sets 
and 
there is a set whose only elements are and . (Note that taking 
, this gives you the possibility of forming the singleton set 
.)
Axiom 4: For any set 
there's a set whose elements are the elements of the elements of A. (If 
then this set is what's called the union 
of 
and 
, whose elements are all the elements of together with the elements of . For example if 
and 
then 
.)
Axiom 5: For any set 
there is a set, called the power set of , whose elements are the subsets of 
(A subset of is a set whose elements are all contained in . Power sets are an important ingredient of Cantor's hierarchy of infinities.)
Axiom 6: For any function 
defined on a set 
, the values 
where 
is an element of 
also form a set.
Axiom 7: Every non-empty set 
has a member whose elements are all different from the elements of . (This axiom insures that if you take an element 
of and then an element 
of , and so on, this process stops after finitely many steps.)
Axiom 8: There is an infinite set such that if 
is in this set, the set 
is also in the set.