Can prosecutors present evidence of a defendant's bad character?
Evidence that’s too prejudicial—that is, too likely to lead to unfair bias against the defendant—is generally inadmissible. Different states have different rules, but in general, evidence that is more prejudicial than probative can't come in at trial. Many courts focus on the risk that the evidence will be too prejudicial, and many say that the potential prejudice has to substantially outweigh the probative value.
This prejudicial/probative question created the framework for the prevailling rule on character evidence. According to it, evidence that simply shows that a defendant is likely to have committed a crime because of her character isn't admissible.
For example, evidence that a defendant on trial for robbery previously robbed other victims would usually be inadmissible. True, the fact that she previously committed the same crime tends to support her guilt on the occasion in question. But, regardless of the other evidence against her, it’s also likely to convince the jury that she’s guilty this time around—or that she deserves a conviction for simply being a bad person. (For a real-life example of evidence that's too prejudicial, see New Jersey High Court Distinguishes Rap Lyrics From Evidence of Guilt.)
Inadmissible character evidence can consist not only of evidence of the defendant's previous misdeeds, but also of evidence of her reputation or of someone's opinion of her. (For further explanation, see Defendants and Character Evidence.)
There are, of course, exceptions to the rule against character evidence. Cases involving domestic violence and sex crimes provide examples. (See Defendants and Character Evidence.)