From Wikipedia
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems.
An axiomatic system that is completely described is a special kind of formal system; usually though, the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans.
What do "completely described" and "complete formalisation" mean? Thanks.
Take a look at a mathematical logic textbook, like George Boolos & John Burgess & Richard Jeffrey Computability and Logic (5th ed - 2007); you will find versions of "formalized" arithmetic [see page 214-on] : usually Peano arithmetic ($\sf PA$) or some subset of it, like Robinson arithmetic.
Those system are completely formalized because they are based on the syntax and proof system of first-order logic, i.e.
the language of the system is specified : rules for terms and formulae formation
a complete list of logical axioms and rules of inference
a rigorous definition of what means to be derivable in the system, i.e. a definition of theorem
an "explicit" list of non-logical axioms (the f-o version of Peano axioms).
Having done this, we are able - in principle - the check in a "mechanical" way :
if a string of symbols is a (well-formed) term or formula of the system
if a formula is an (instance of an) axiom (logical or not) of the system
if a sequence of formulae is a derivation, in which case the last formula of the sequence is a theorem.
But, and this is the real value of this approach, we can study the system itself mathematically; i.e. we can prove relevant "facts" about the system, "doing" metamathematics : consistency, completeness, relation with weaker and stronger systems, and so on...