In First Order Logic i just studied Axioms' Schemes, that are:
An axiom schema is a sentential formula representing infinitely many axioms. These axioms are obtained by replacing variables in the schema by any formula.
The axioms obtained are the same thing of formulas (in First Order Logic) or are different things ? If they are differents, how i obtain formulas from axiom schemes or from axioms ?
For propositional logic we have:
Language: connectives and sentential variables: $p_1,p_2, \ldots$.
Formulas are expressione formed with variables and connectives according to the formation rules; e.g. $\lnot p_1, p_1 \to p_2$ are example of formulas of propostional calculus.
A schema is an expression of the meta-language where the (meta-)variables: $\varphi, \psi,\ldots$ stay for formulas.
An axiom schema is an expression of the meta-language, like:
and it must be read as a "recipe" to generate infinitely many axioms (called instances of the schema).
How ? Replacing uniformly the meta-variables with formulas of the language.
Thus, form the axiom schema: $\varphi \lor \lnot \varphi$ we can generate the axioms:
The same for predicate calculus (with the obvious changes regarding the basic elements of the language and the formation rules).
An example of axiom schema can be $\forall x \alpha \to \alpha[t/x]$ and a corresponding instance is: $\forall x (x \ge 0) \to (1 \ge 0)$.
In conclusiom: axioms are expression of the (formal) language; axiom schema are expression of the meta-language.