Note: My answers are based primarily on the books themselves and do not use many details (or lack thereof) from the movie.
Azkaban is/was the most feared prison in the London side of the wizarding world. This is because of the cold dark cells combined with the Dementor guards. During the time that Voldemort was left powerless, the Dementors had nowhere to go. So the Ministry employed them into Azkaban in order to keep them under control in a sense and to further discourage wizarding crimes.
The main reason no-one escaped from Azkaban ever was because the power of the dementors left the prisoners in such a state of apathy that they felt escape was pointless and even that wouldn't bring happiness back into their minds.
The ultimate punishment for an offending wizard was not to be sent to Azkaban as a prisoner, but to have performed upon them what became to be known as the Dementor's Kiss. This act would allow the Dementors to unveil their hood and suck out the soul of the offending wizard, leaving them as an emotionless husk for the rest of their days. A fate considered much worse than death or anything Voldemort could accomplish during his reign of terror.
In the 3rd book, the dementors didn't escape. They were merely allowed to hunt down the only escapee to ever come out of Azkaban: Sirius Black. This act was only performed by Siris because he was an unregistered Animagus someone who could turn into an animal at will, but did not register this ability of his with the Ministry of Magic. Without that information, the Ministry had no clue how he could escape and unleashed the Dementors on the world due to essentially panic on their part.
Sirius figured out this trick due to noting the Dementors had no actual eyes to see with, they merely felt their way around using people's emotions. For whatever reason, they are unable to do this with animals so by turning into a dog, Siris was able to literally walk right out of Azkaban.
Now, jump ahead to the 6th with people being broken out of Azkaban. The Dementors themselves are only magical creatures but they themselves are not capable of magic as humans are. Therefore, it must have been Death Eaters themselves that either blew a hole into the prison using basic spells or even the Dementors decided to stop feeding on the imprisoned Death Eaters and allow them to simply leave the prison due to Voldemort's return and guarantee of fresh food.