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" I understand how the engines work now. It came to me in a dream. The engines don't move the ship at all. The ship stays where it is and the engines move the universe around it."
Cubert Farnsworth[source]

The Dark matter engine is the key to the extreme acceleration capabilities of the modern space ship. Invented by Professor Hubert J. Farnsworth, the engines on the Planet Express ship harness the power created by burning dark matter in large furnaces, channels it through an afterburner that gives 200% fuel efficiency and propelles the ship through space fast enough to cover the whole universe in a matter of days.

How it works

Let's look at the real universe example of Rigel, a star in Orion's Belt approximately 900 light-years away from Earth. This means that even traveling at the speed of light (300,000 km/s) it would take 900 years to get there. Traveling at 9 times the speed of light (2,700,000 km/s) it would take 100 years to get there and at 100 times the speed of light (30,000,000 km/s) it would take nine years. Albert Einstein's famous statements that it would be mathematically impossible to travel faster than light seem to have held up throughout the 3rd millennia and although in 2208 the scientific community allegedly raised the speed of light so that they could go faster, the problem that one cannot go faster than the speed of light remains.

The solution was the dark matter engine, which doesn't move the ship through the universe, but instead moves the universe around it at phenomenal speeds and is thereby able to cover incredible distances in a relatively short period of time.

Real Life?

Some time in the last 10 or so years, such a mechanism has been seriously proposed by physicists. That is, rather than moving through space at superluminal speeds, the drive "bunches up" the desired amount of space into something the length of the ship, which then hops through quickly and allows the bunched-up space to relax to normal size (now behind the ship). There was no mention of Futurama, which either means that the physicists were too embarrassed to mention that they got the idea from a sci-fi sitcom, or they came up with it independently. Of course, there was no word on just how you would bunch up space in such a manner -- probably more "negative mass" handwaving than solid theory. If anyone can find the published material on this proposal, a link to it would be nice. The proposal was called the EmDrive, this stands for Electromagnetic Drive and the paper is found here.

Also, fun fact, this is how a lot of the old 2D video games worked. The images which made up the "map" were moved in the opposite direction that player "travelled" to give the illusion of movement.