In the 1970's it was discovered that stars in spiral galaxies rotate around the center in a very unusual way. The stars in the outer parts of the galaxies seem to be moving too fast for the galaxy to hold them on circular orbits - they ought to fly off into intergalactic space. Since galaxies are common and stable enough looking things, it seems unlikely that we would be lucky enough to be getting a look at them through our telescopes moments before they fly apart and dissolve into the depths of space.
It has long been known that galaxies spin around their center, much like the planets orbit around the sun. And like planets orbiting the sun, they follow Kepler's Laws for orbits around the center. These laws state that the rotational velocity around the center depends only on the distance to the center, and the total mass that is contained within the orbit.
So, by finding the rotation velocities along a galaxy, one can `weigh' the mass of the galaxy inside that orbit. Since as you go along the edge of a galaxy, the amount of light quickly starts falling off, one would expect the rotation speeds to fall off, too; but they don't. Instead, the rotation speeds remain high above what one would expect - which indicates strongly that there is a great deal of mass in the galaxy that we can't see. This has been done for many spiral galaxies - galaxies like our own - with the same results, and is the first and strongest evidence for dark matter within galaxies.
Here is a graph to illustrate the difference between expected rotational speeds and observed rotational speeds at various distances:
From this one can infer that the distribution of mass in the galaxy can not be concentrated similar to the light distribution, that is the mass of the galaxy must continue to increase as you move out from the center of the galaxy.
It is estimated that as much as 90% of matter in galaxies is "dark matter".
What is "dark matter"? Black Holes? Neutrinos? Brown Dwarfs? Exotic Particles? Other?
Finally, a look at the universe around us will give a hint of more structure than meets the eye. The graphic I've linked will show the "soap bubble" or "sponge" appearance of the superclusters and voids. Enjoy!