Relevance is the basic building block of evidence rules—evidence must be relevant to be admissible. For evidence to be relevant, there must be some logical connection between it and the fact it's offered to prove or disprove. The connection needn't be so strong that any single item of evidence alone proves or disproves the fact. It's good enough if the piece of evidence constitutes a link in a chain of proof. (As the famous legal authority McCormick put it long ago, "A brick is not a wall.")
Example: Ruby Ridge is charged with stealing costume makeup from a drugstore the night before Halloween. The prosecution wants to offer evidence that Ruby's mom had refused to buy her a Halloween costume. The evidence is relevant to prove that Ruby had a motive for stealing the makeup.
Example: Same case. The prosecution also wants to call the drugstore manager to testify that the makeup department suffers more thefts than any other department of the drugstore. This testimony would be irrelevant because it does not relate specifically to Ruby.
Example: Lance Sellot is charged with drunk driving. The prosecution wants to offer evidence that Lance is a member of a violent street gang. The evidence is irrelevant because the crime charged has nothing to do with gang activities. The evidence would only serve to stir up bias against Lance.
Example: Clare Voyant is charged with car theft. She was arrested in her home, and the prosecution wants to offer evidence that the arresting officer found marijuana and an unregistered handgun there. Unless the prosecution can establish that the gun and drugs were somehow involved in the theft, there's nothing to connect them with the crime. Again, the evidence would do almost nothing other than predispose the judge or jury against the defendant.
Evidence has to be relevant to have any chance of admissibility, but not all relevant evidence is admissible. Judges often exclude relevant evidence because of some other evidence rule. For example, evidence that is relevant may be likely to unfairly arouse the jury's emotions. In such situations, the judge is supposed to balance the importance of the evidence against the risk of an unfair appeal to emotion. If the judge determines that the risk of unfair emotional appeal substantially outweighs the relevance, the evidence won't come in. Otherwise, it will.
Example: Kai Ping is charged with assaulting Kevin Pong with a knife; Ping claims self-defense. The prosecution seeks to offer into evidence (a) the knife allegedly used in the assault, (b) a photograph of Pong taken minutes after the fight, showing cuts on his face and arms, and (c) the blood-soaked T-shirt that Pong was wearing at the time of the fight. A judge is likely to admit the knife and the photograph into evidence, but might exclude the shirt as unduly prejudicial. The size and shape of the knife and the nature of Pong's injuries are rationally related to the issue of whether Ping or Pong was the aggressor in the fight. But, if the shirt doesn't provide much information about how the events unfolded, the judge will likely exclude it because of its potential to shock and disgust the jury.