Rule of Inference: Statement to the effect that if one has asserted (said, written) one or more of certain kinds of statements, collectively called the premisses, one may, therefore, assert (say, write) some kind of statement, called the conclusion.
Human beings use and rely on many kinds of rules of inference in their speaking, writing and argueing, but often  when not much concerned with logic  are not much aware of this, and anyway would find it difficult to precisely state the rules of inference they use.
Here is an example of a rule of inference for the term "and", when used as a term that connects two statements into a statement, that is used and relied upon by the vast majority of English speakers: "If p and q are two arbitrary statements, and one has asserted that p and q is true, then one may assert that p is true".
Any rule of inference is valid or invalid, and is so with respect to some interpretation of the terms in the rule, that state what the terms mean. A rule of inference is valid iff its conclusion is true in every case that all its premisses are true. Consequentely, a valid rule of inference has the property that, for the given interpretation, it can conclude only true statements from true premisses.
A rule of inference is invalid iff it is not valid.
In general in logic and mathematics, deductively valid inferences correspond to valid implications, but do not reduce to the latter, in as much as they add a permission to infer (write, assert) a conclusion if it is logically implied by earlier proved or assumed formulas.
