The prosecution of a defendant for an offense for which e has already been tried... usually. On the other hand, the phrase 'double jeopardy' is often used to refer to the laws forbidding prosecution for a crime that the defendant has already been acquitted for. That’s right, 'double jeopardy' is a contronym. Double jeopardy (in the first sense) is banned under the legal systems of many countries.

In the United States of America

The "common-law and constitutional (Fifth Amendment) prohibition against a second prosecution after a first trial for the same offense. The evil sought to be avoided is double trial and double conviction, not necessarily double punishment." – Black's Law Dictionary 440 (5th ed. 1979).

This doesn’t apply across the board tho; there are at least two important exceptions.

Multiple jurisdictions:
In United States v. Lanza, the Supreme Court ruled that double jeopardy does not preclude federal and state prosecution for the same offence. "An act denounced as a crime by both national and state sovereignties is an offense against the peace and dignity of both and may be punished by each." – Supreme Court case: United States v. Lanza, 43 S. Ct. 141. (1922) This was later extended to allow for double jeopardy to occur when two different states were bringing the charges (Heath v. Alabama, 106 S. Ct. 433, 437-438 {1985}). **

Civil V. Criminal charges:
In Barnes v. Tofany the Court stated "the constitutional prohibitions against double jeopardy and double punishment do not prevent the legislature from enacting, and the executive from enforcing, civil as well as criminal sanctions for the same conduct." -- Matter of Barnes v. Tofany, 27 NY2d 74 (1970). (This is probably not the earliest case to support this decision, but it is the earliest I have found).

On the other hand…
In the case of United States v. Halper, the Court decided that double jeopardy would be denied in a civil case, because the criminal case had already punished the defendant enough. "{T}he labels 'criminal' and 'civil' are not of paramount importance"; it is more important that the punishment fit the crime.

"{T}he Double Jeopardy Clause protects against three distinct abuses: 1 a second prosecution for the same offense after acquittal; 2 a second prosecution for the same offense after conviction; and 3 multiple punishments for the same offense." --United States v. Halper, 490 U.S. 435 (1989).

"…nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb…"
-- Amendment V, The Constitution of the United States of America.

About a year ago, in the UK, it was suggested that protection from double jeopardy should be eliminated, at least in some cases. In serious cases such as murder, a defendant should be able to be retried if new evidence came forth*.

This brought an obvious question to the minds of many: why do we have these laws in the first place? Well, as stated above, this is a codification of a common law tradition, but there are other reasons.

Double jeopardy could open the door for politically motivated prosecutions. The government (any government) is much more powerful than is its subjects, and the ability to repeatedly prosecute would be a powerful weapon against anyone – even an innocent person. Even if you trust the government, you should not trust everyone who holds a position of power in the government. (Along with this goes the belief that since the government is calling all the shots, they should get all their ducks in a row before bringing a case to court. If they don’t, that’s their own fault.)

The fact that a person has been tried once might sully the presumption of innocence. When a jury is told that the defendant has been tried once before, they are more likely to assume that e is guilty. This is an obvious problem for the defendant.

* The UK currently does not allow double jeopardy.

** The UK (unlike the USA) recognises the principle of autrefois acquit, so that someone tried (satisfactorily) in another jurisdiction may not be tried again for the same crime in the UK.
Thanks to texture for these footnotes.