This account follows the definition of "V" (i.e. the collection of sets which form the iterative or cumulative heirarchy) given in lecture notes (728Kb postscript) for a course on The Foundations of Mathematics by Stephen G. Simpson.
I believe this is pretty much the standard story, except perhaps for the question of what an ordinal is. Ordinals are perhaps more often defined as certain sets (the Von Neumann ordinals) rather than as isomorphism classes. 
This all hinges around two things.
The first is the definition of an ordinal.
The second is the power set constructor.
The stages in the iterative construction are the ordinals and at each stage you get the union of the power sets of all the previous stages.
On top of any problems we may have with the ordinals and the power set constructor, there is added at the end the question whether it makes sense to suppose this process can ever be completed. At some stage I will probably rewrite this with a different definition of the ordinals. 
A relational structure is a pair (A,R) where R (AA) 
Two relational structures (A,R) and (B,S) are isomorphic if:  

There exists a bijection between A and B which respects (preserves) the relations R and S, i.e. maps pairs in A related under R to pairs in B related under S and viceversa.  
The isomorphism type of a relation is the class of relations which are isomorphic to it. 
A relational structure (A,R) is wellfounded if:  

For every subset X of A there exists an element a which contains no element of X (a is disjoint from X). 
A relation (A,R) is transitive if for any three elements a,b,c in A, if a R b and b R c then a R c.
A relation (A,R) is a linearordering if for any two elements a,b in A eactly one of a R b, a=b and b R a holds. A wellordering is a linearordering which is wellfounded. 
An ordinal number is the isomorphismtype of a wellordering. 
A set is transitive if all its members are also subsets.
The transitive closure of a set A (TC(A)) is the smallest superset of A which is transitively closed. A set A is wellfounded if the relational structure (A, {<x,y> (AA)  x y}) is wellfounded. A set is pure if every element of its transitive closure is a set. 
We define corresponding to each ordinal number a stage V() in the construction of the cumulative hierarchy as follows:
The iterative or cumulative hierarchy is the union of the V() for every ordinal . 