Dark matter is Magic. Not only is it undetectable (hence the name dark), it's also incredibly common — 10 times more so than ordinary matter — and miraculously solves any equations it's faced with. Why is this?
Because we made it up. Dark matter, and its even more slippery counterpart, dark energy, is a big plaster on some massive gaps between our theories and our observations of the universe.
Dark matter was first co-opted to persuade galaxies to hold together. To prevent a galaxy breaking up as it spins, gravity must provide the force needed to stop the stars flying apart. However, the strength of the gravitational force depends on the mass of galaxy: and when you look at individual galaxies, you see that adding up the weight of all the stars doesn't make the galaxy heavy enough to keep the stars orbiting round as fast as they are. This is a Problem.
There are a few choices as to how to deal with this:
No prizes for guessing which one they went for on this occasion, and given how successful gravity and centripetal force are (consistently making the top ten for 'The Nation's Favourite Forces') it was a fair assumption to make. But in science, there is somewhat of an embargo on spontaneously inventing unobservable things that miraculously make your equations fit the experiments. It is, in a word, cheating.
But rather irritatingly, cheating seems to work. An example: When Newton first came out with his theory of gravitation, it got the orbit of Uranus wrong. Rather than cast aspersions on Newton's shiny new theory, (or upset Newton, who was a bit of a curmudgeon) it was suggested that perhaps there was another, as yet undiscovered planet, which was affecting the orbit in the way that Newton predicted. When they looked for this planet, they found Neptune — a new planet, as requested, which made Uranus orbit in the way Newton thought it ought. So in this case, clinging to your theory and inventing convenient things worked out. But it's pretty easy to test if a whacking great planet exists, so perhaps it's not quite the same.
A better comparison is with Pauli's cunning invention of the neutrino. When observing elementary particles before and after collisions, a certain amount of energy seemed to be lost. In order to keep energy conserved, Pauli hypothesized a neutral, very light, undetectable particle called the neutrino which would carry away some kinetic energy each time, and thus make the equations balance and yet not be visible. Another invisible fudging device — but, in 1956, some intrepid men with a very large experiment discovered the neutrino. Again, Pauli's gratuitous invention had been on the money.
So what's my problem with positing this stuff? It seems to work.
Well, firstly, it's not always successful. After the trouble with Uranus, it was pointed out that Mercury doesn't orbit as predicted either. People tried the same technique again, inserting a planet called Vulcan between Mercury and the Sun, but Vulcan refused obstinately to appear in the telescopes. Hardly surprising, because it didn't exist: the orbit of Mercury is screwy because of the way massive objects like the Sun bend space and time, which wasn't elucidated until Einstein did his thing in 1915.
More importantly, though, it's just a bad idea. It's too easy to fall into the trap of giving something a name and assuming you know more about it now (think of the people who would feel they knew more about a light in the sky if they were told it was an Unidentified Flying Object. They might even feel like they knew something other people didn't!). Covering our ignorance by naming it 'dark matter' and then defining it according to what would make your equations work is shaping the world after the scientific theory, and it really ought to be the other way round. After all, there are lots of scientific theories, but only one world.
Dark matter is a vast tract of human ignorance. If it exists, the content of the vast majority of the universe is a mystery to us; we can make maps of it using indirect effects, but we are no closer to knowing what it is made of or what it does. If it doesn't exist, one of our big theories — basic mechanics, or Einsteinian gravitation, is wrong, and the motion of the universe is again a closed book. It's rather humbling how little we know. And we're not learning any more by thinking about what miracles we'd like this hypothetical material to perform, or what theoretical holes we'd like to plug.
Of course, dark matter may be valid; it might be a new type of particle or a concept we're just not familiar with yet, but that still leaves all our current laws of nature unchanged (not that any other radical scientific discovery has left all previous work intact, but it could happen... I guess...). In that case, even more reason not to assume anything, to just take the (scanty) evidence so far, acknowledge the gaps and fill them in with experiments rather than wishful thinking.