How do we know there's dark matter in the universe?
Well, the reason we even suspect it's out there is because according to Newtonian physics galaxies should fly apart under their own momentum. Yet it doesn't happen, so we have to assume there's some strange dark matter keeping it together with some extra gravity.
Currently, there are two reasons why we believe in dark matter, and the evidence like suggested above comes from:
1) In spiral galaxies
2) In clusters of galaxies
Found these crazy explanations on another site

- In spiral galaxies like the Milky Way, we derive the gravitational mass from observing the motions of stars and gas clouds in the disk as they orbit the center. The rotation curve of a galaxy shows how the velocity of stars around the center varies as the distance from the center increases. Most spiral galaxies show flat rotation curves out as far as we can trace them, even where no more stars are visible. Therefore we conclude that the gravitational mass is more than 10 times more massive than the luminous mass.
- In clusters of galaxies, we derive the gravitational mass by measuring the orbital motions of the member galaxies. Since the galaxies in a cluster are roughly at the same distance from us, we can interpret any spread in their redshifts as orbital motion around the center of the cluster; it might amount to more than 1000 km/sec! By measuring the redshifts of lots of galaxies in the cluster, we can calculate the gravitational mass required to keep the galaxies in orbit (rather than escaping). This gravitational mass then can be compared to the luminous mass contributed by the galaxies plus that contributed by the Xray gas.