Key features include: definitions, postulates and common notions as starting points from which the propositions of the subject are derived. Key defects are the lack of formality, the intuitive nature of derivation, and hence the danger of hidden premises. 
Logical axioms correspond to common notions, nonlogical axioms correspond to postulates, and primitive definitions to undefined logical constants. 
The foundational approach involves a strict axiomatic treatment of the theory of sets, which establishes an ontological framework within which all other mathematical theories can be introduced by definitions rather than axioms. The benefits of this are:

Because of the risk of inconsistency highlighted by paradoxes such as Russell's paradox, not all the sets one might wish to see are available in classical set theory. It turns out that as mathematics becomes more abstract many of the structures which one might wish to reason about just don't exist in this context. For example, mathematical groups are related among themselves by homomorphisms, and we may wish to consider the structure of this network of homomorphisms and relate this structure to others, perhaps the homomorphisms between fields. These collections of homomorphisms are examples of another kind of structure called a category. Unfortunately, the collection of group homomorphisms in ZermeloFraenkel set theory is not itself a set, and so the categories we are interested in relating simply don't exist in this context. 