You need a textbook of mathematical logic ...
See Sara Negri & Jan von Plato, Structural Proof Theory (2001), page 4.
First of all, we have a formal language, with rules for building expression (terms and formulas).
Then we need rules of inference; they are of the form:
"If it is the case that $A$ and $B$, then it is the case that $C$"
where $A, B$ and $C$ are formulas of the language. Rules allows us to move from given formulas to new ones.
In Hilbert-style systems, (also called axiomatic systems), we have a number of basic forms of assertion (axioms), such as $A \rightarrow (A \lor B)$ and at least one rule of inference (tipically : modus ponens).
In natural deduction systems, there are only rules of inference, plus assumptions
to get derivations started.
With these "ingredients", we may use our proof system : we call theorems of the proof system all the formulas that are deducible, i.e. “produced” starting from the axioms with a finite number of applications of the rules of inference.
The basic properties we are requesting to it are the following :
1) it must be sound : assuming a model for our axioms (so that they are true in it), we want that all the theorems must be true in that model (i.e. rules of inference must preserve truth);
2) it must be complete : we want that all the logical consequences of our axioms must be deducible from the axioms.
