If we have two axioms $A$ and $B$, what exactly is meant by axiom $A$ being weaker than axiom $B$? This question is a follow-up to A weaker Axiom of Infinity?
We are working within a certain theory $T$ and we are contemplating adding Axiom $A$ or Axiom $B$. Then (in the context of $T$) Axiom $A$ is (strictly) weaker than $B$ if from $T$ together with $B$ we can derive $A$, but from $T$ together with $A$ we cannot derive $B$. Often there is a tacit assumption that $T$ is consistent.
Axiom A is said to be weaker than axiom B if axiom B implies axiom A.