You know Dutch angles even if you don't know that you know, you know?
These shots have appeared in everything from The Man with the Movie Camera to Casino to The Avengers. They are used by brilliant filmmakers to amazing effect and by hackneyed filmmakers to, well, hackish effects. (Seriously, Battlefield Earth, how are you a thing?)
A Dutch angle is simply a shot where the camera is titled at an angle rather than showing the scene straight on. This is why it is also called the canted angle or oblique angle. The degree of the tilt can be anywhere from 10 to 70, and one shot in 2001: A Space Odysseyfeatured a 90-degree tilt (source).
What are they used for? That really depends on the movie, but generally speaking, a Dutch angle creates a jarring effect designed to contrast to the rest of the film's straight shots. This can be used to set an eerie mood in a horror movie, a suspenseful build-up in a thriller, or a twist of fate in an adventure movie.
In 12 Monkeys, the Dutch angle is used to show us that things have gone askew. For an example, we'll consider when Cole first goes back to 1990. The scenes where he's diagnosed by Dr. Railly in the prison cell are shot with a straight angle. But once he gets to the mental hospital, the Dutch angles come out to play. Almost the entire scene where Goines introduces Cole to the hospital's rules is shot at a Dutch angle, showing us that Cole's understanding of the world has gone awry. It also allows the audience to experience that sensation with him.
Other examples include Cole's talk with Goines at the dinner and Railly's discovery of the WWI picture featuring Cole in the background. Both of these scenes prominently feature Dutch angles to show how the characters' perceptions of the world deviate from the norm.
Oh, and here's a fun fact: the Dutch angle isn't Dutch at all. The shot was introduced by German Expressionists, and it was called the "Deutch angle." Because, you know, "Deutch" means "German" in German. But then people got lazy or confused or whatever, and Holland got all the props.
Meanwhile, Germany has to settle with the knowledge that its artists have created timeless masterworks like The Cabinet of Dr. Caligari. Actually, that's not a bad trade-off (source).