As-applied challenge

When challenging the constitutionality of a law, the normal method of seeking invalidation is a facial challenge, which invalidates the law as having no constitutional applications whatsoever. Most Supreme Court cases are resolved in this manner.

However, sometimes a facial challenge is untenable, or difficult to argue. The easier argument to win, often, will be an as-applied challenge, seeking a declaration that the selected law is invalid as applied to your client, based on the particular facts of the scenario.

Examples
The best example of a facial vs. as-applied challenge in recent memory is the case of In this case, the Federal Government, acting as a patron of the arts through the NEA, promulgated guidelines for NEA grants that granted the NEA the power to evaluate works subjectively before funding them, on the basis of the offensiveness of the work in question.

Content-specific criteria like these are generally invalid when they function as a restraint on free speech. Normally, the interest of the speaker trumps the interest of the government in restraint. However, the Finley case posed a problem of competing interests: after all, NEA funding is limited, so the government ought to have the ability to make subjective decisions, especially given the subjective nature of art. Therefore, the Finley court held that, while the NEA's subjective judgment criteria were not facially unconstitutional — that is, they could be applied in a constitutional manner, in line with legitimate government concerns — they could be unconstitutional as applied. For example, where a denial of an NEA grant occurred merely because the art in question would have depicted homosexuality, the NEA criteria would presumably be unconstitutional.

The idea is that, if a statute is legitimate most of the time, it generally shouldn't be struck down, even though it has unconstitutional outcomes at the extreme.

Usage by the Roberts Court
The Supreme Court under Chief Justice Roberts has tended to highlight the differing uses of facial versus as-applied challenges.
 * In Gonzales v. Carhart, Justice Kennedy declined a request to invalidate a law that would have banned an abortion procedure — specifically, dilation & extraction — even when the health of the mother required as much. Kennedy writes that the procedure is only beneficial at the extremes (making the law only unconstitutional at the extremes), and therefore the law should be vulnerable to an as-applied challenge, but not a facial challenge.
 * In the Supreme Court considered the  (BCFRA), which, in part, prohibited issue advertisements within 30 days of an election. Chief Justice Roberts invalidated the act "as applied" to Wisconsin Right to Life's particular issue ad, saying that this prong of the Act would rarely be constitutional, and closed his opinion by expressing his belief that the entire Act was unconstitutional.

The special case of free speech
Some speech regulations may be challenged facially even though they would be constitutional as-applied to the challenger. This is an exception to the general rule that a statute may only be challenged (facially or as applied) if it infringes directly upon your rights.

For example, suppose a writer who publishes a book calling for the assassination of George W. Bush is prosecuted under a statute which prohibits criticizing George W. Bush. Although the writer could be punished for his own acts, the fact of the statute's overbreadth makes it facially invalid, and allows the defendant to walk freely. This special leeway is afforded in speech cases to prevent federal or state governments from drawing statutes deliberately to "chill" speech.