If the axioms are a first-order theory then we have Goedel's Completeness Theorem which states that a theory is consistent if and only if there is a model of this theory.
So one theory proves another is consistent if it can show the existence of a model of the second theory. For example, in ZFC we can show that the finite ordinals form a model of PA, therefore as a first-order theory, PA is consistent if ZFC is consistent.
Of course, this says nothing about whether or not PA is really consistent or if ZFC is really consistent, this boils down to "believing" the axioms you work with are true. However we do know that granted ZFC is consistent then we can generate a model of PA which then shows that PA is consistent as well (this is called relative consistency, we say that PA is consistent relative to ZFC).
The Incompleteness Theorem prevents ZFC from proving its own consistency and for that we need to have an additional axiom, giving us a stronger theory which can then prove ZFC is consistent (such axioms are "ZFC is consistent", or "There exists an inaccessible cardinal", etc.)