Prime among the crimes for which someone can be sent to Azkaban is the performing of one of the three Unforgivable Curses: the Imperius Curse, the Cruciatus Curse and Avada Kedavra. Perfoming one of these curses against any human is an automatic life sentence at Azkaban.
Dementors were chosen to guard Azkaban since they are extremely difficult to injure - the only charm useful against them, Expecto patronum, only repels them. Also, they sap the happiness out of people, and hence the wizards under imprisonment are gradually rendered helpless. In case of an emergency, they can also perfom the Dementor's Kiss, where they suck the victim's soul out of his mouth.
In the second book, Hagrid is placed under temporary imprisonment at Azkaban - an experience so traumatic that he refuses to talk about it after being released. Until Sirius Black becomes the first wizard known to break out of Azkaban (the first one to break out of this place in the Harry Potter universe was Barty Crouch Jr) in the third book, Azkaban was believed to be escape-proof.
Professor Dumbledore has always been vocal in declaring that it was a mistake to guard Voldemort's greatest supporters with those creatures who have the most to gain if Voldemort returned to power. In the fifth book, this is in fact what happens.