Here are some examples of axiom schemas that are valid in modal logic :
T refers to the axiom scheme $\square\mathrm A\rightarrow \mathrm A$
5 refers to the axiom scheme $\diamondsuit\mathrm A\rightarrow \square\diamondsuit\mathrm A$
K refers to the axiom scheme $\square(\mathrm A\rightarrow\mathrm B)\rightarrow (\square\mathrm A\rightarrow\square\mathrm B)$
Df♢ refers to the axiom scheme $\diamondsuit \mathrm A\leftrightarrow \neg\square\neg\mathrm A$
which can be interpreted informally as follows:
If $\mathrm A$ is necessary, then it is true.
If $\mathrm A$ is possible, then it is necessarily possible.
If it is necessary that $\mathrm B$ be true whenever $\mathrm A$ is true, and if $\mathrm A$ is necessary, then $\mathrm B$ is necessary.
$\mathrm A$ is possible if and only if it is not necessarily false.