Evidence is relevant when it has any tendency in reason to make the fact that it is offered to prove or disprove either more or less probable. To be relevant, a particular item of evidence need not make the fact for which it is offered certain, or even more probable than not. All that is required is that it have some tendency to increase the likelihood of the fact for which it is offered. Weighing the evidence is for the finder of fact, and although a particular piece of evidence, standing by itself, may be weak, it will be admitted unless it is otherwise incompetent or it runs afoul of an exclusionary rule. For example, if the fact to be proved is that the defendant bit off the plaintiff's nose in a fight, testimony by an eyewitness to the act would clearly be relevant, but so would testimony by a witness who heard the plaintiff and the defendant exchange angry words on the day before the fight, or even testimony by a witness who sold the defendant a disinfectant mouthwash shortly afterwards.
Evidence is material if it is offered to prove a fact that is at issue in the case.
For example, if I offer the testimony of an eyewitness to prove that it was raining on the day of the signing of a contract, that evidence may be relevant to prove the fact for which it is offered, yet the fact that it was or was not raining may be immaterial to any of the issues in the case, which may turn entirely on whether one or both parties breached the contract. The issues in the case are determined by the pleadings, any formal stipulations or admissions, and the applicable law. For example, if, in a case of breach of contract, the defendant has conceded that the plaintiff performed all his covenants, proof of that performance would no longer be material unless it were relevant to some other issue. Under both the California and federal rules, the concept of materiality is included in the concept of relevance. Evidence is competent if the proof that is being offered meets certain traditional requirements of reliability.