When is a law overbroad?
A law is overbroad if it substantially prohibits conduct protected by the Constitution, such as forms of protected speech. For example, in a recent case, United States v. Stevens, the Supreme Court found that a law banning the commercial creation, sale, or possession of depictions of an animal being intentionally and illegally mutilated, tortured, wounded or killed was unconstitutional because it was overbroad.
Among other things, the Court found that the law could, in some cases, criminalize photographs of hunting and livestock. Although the law constitutionally prohibited certain types of speech, the statute would also substantially apply to protected speech, too.
Similarly, a law that criminalizes any depiction of an unclothed person would be overbroad. Although the law might constitutionally prohibit some types of pornographic material it would also criminalize some forms of art, news footage, etc.
When is a law too vague?
Vagueness is slightly different but can easily overlap with a statute that is overbroad. A statute is too vague if people of common intelligence would have to guess at the statute’s meaning. For example, in Coates v. Cincinnati,the US Supreme Court held that a statute which criminalized three or more people congregating on a sidewalk and “annoying” other people was too vague. People would have to guess as to what conduct law enforcement would consider “annoying”. By forcing people to guess what the statute meant, the law violated the Constitutional right to due process of law
The law was also overbroad because although it applied to behavior that a state can constitutionally prohibit, such as obstructing a sidewalk, the language of the law would also criminalize Constitutionally protected conduct.