"I'd blow meself up before I set foot in that place...Them Azkaban guards give me the collywobbles."
     -- Ernie Prang

"They don't need walls and water to keep the prisoners in, not when they're trapped inside their own heads, incapable of a single cheerful thought. Most go mad within weeks."
     -- Remus Lupin

The Department of Magical Law Enforcement maintains a prison on a tiny island far out in the cold north of the North Sea (Sch1) [see note below]. It's called Azkaban Fortress or simply Azkaban.

Azkaban is a terrible place. The prisoners there are guarded by Dementors, horrible creatures who have been described as "souless fiends." They feed on positive emotions, which means that after spending much time in the prison, the prisoners lose all hope and all good feelings and thoughts. They prisoners are forced to relive the darkest, most horrible memories of their lives. Some go mad, others simply despair and die. (PA10) 

Barty Crouch Sr. was head of this department during the Voldemort years (GF27) and he prosecuted many suspected Death Eaters and other supporters of Voldemort. As the struggle with the Dark forces became more and more desperate, Crouch advocated very strict countermeasures, until the Aurors under his command used methods not much different from those of the wizards they were trying to capture. Crouch sent people to Azkaban without proper trials; Sirius Black, for one, was sent there even though he was innocent. Many Death Eaters are imprisoned there under the watchful eye of the Dementors.

The Council of Magical Law may possibly be located at Azkaban, since prisoners are escorted and guarded by groups of Dementors (GF30) [more likely it's in London]. 

Known prisoners in Azkaban, past and present:

• Sirius Black, falsely convicted and sent c. 1981,  escaped late July, 1993
• Barty Crouch Jr., escaped disguised as his mother, who died there in his place c. 1981
• Karkaroff, released after cooperating with the Ministry by naming other Death Eaters, c. 1981
• Rubeus Hagrid, held there for a few weeks in the spring of 1993
• the Lestranges, married couple, possibly for Cruciatus assault on the Longbottoms, still incarcerated
• Anton Dolohof, sentenced c. 1981
• Evan Rosier, captured c. 1981, possibly sent to Azkaban. Deceased.
• Wilkes, captured c. 1981, possibly sent to Azkaban. Deceased.

NOTE ON LOCATION:
Because of a comment by Sirius Black describing his escape from Azkaban, some have conjectured that the island must really located farther south. Sirius does state quite clearly that he swam to shore, then visited Privet Drive before heading north to Hogwarts. Privet Drive is in Surrey, which is south of London. Since it is only logical to assume the Sirius would have swum to the nearest point on the mainland to Azkaban, that point must have been somewhere in the English Channel between England and France. Several fans have suggested excellent alternative locations which fit the bill.

This is certainly possible. However, several additional points must be considered. First of all, Sirius was not totally rational before, during, or immediately after his escape. In fact, we do not see him really recovered until he turns up near Hogsmeade a year later. So he may simply be incorrect in what he says, mixing up his directions or leaving out part of his journey. Second, Wizard ideas of "shortest route" don't necessarily use the kind of geometry we Muggle take for granted. The Knight Bus, for example, seems to move between destinations almost in alphabetical order. This odd logic is similar to that which makes transfiguration simpler when the two items have similar names (guinea pigs and guinea fowl). Therefore we can't necessarily take Sirius' offhand analysis of his trip and superimpose it over a Muggle map to figure out where Azkaban is. 

Another way to look at it is this, from Airin on HP4GU: 

"The North sea is not just to the "North" of the UK. It extends most the way down the East coast of Scotland and England. The real question is, where does the English Channel become the North sea? *pulls up a map of the UK to see if this helps* Hmmm. If Azkaban were in a very southerly area of the North sea then Sirius' quick visit would not have been too out of his way, and his journey north would still have been substantial."