Understanding the riddle of Dark Matter requires rewinding the clock all the way back to the Big Bang. Like so many other questions in physics, it can seem odd that two seemingly disconnected topics can end up having such direct bearing on one another. But in the end, unexpected connections like this often end up being a good thing; they are signals, hints that we may have tapped into a fundamental aspect of the Universe that once understood, could help resolve other mysteries as well.
First, what of galaxy structure and rotation? Here the problem is that we cannot detect enough matter to account for the gravity that we know must be present. So, we speculate that something else, something undetectable to us (at least, directly), is producing it.
Perhaps at this point we should quickly address the question of whether some other binding agent, something besides gravity, could be holding galaxies together. Although we are taught never to say never very early in life, it is unlikely to say the least that there could be another force in the Universe grand enough to have these tremendous effects without having been detected before. So, the most reasonable hypothesis seems to be that it is indeed gravity that is holding the galaxies in shape. We'll proceed on this assumption.
The next question then, is where this gravity comes from. In fact, this is the primary question underlying Dark Matter. In the end, the answer to this question may be simple enough to seem almost anticlimactic. Given how perplexing this question has proven to be over the years, it seems that an explanation should be more difficult than it may turn out to be.
First, we must simply understand that according to General Relativity, gravity is not really a force at all; it is a depression in spacetime (more specifically, gravity is a consequence of the curvature of spacetime). To visualize this, we can imagine a tennis ball rolling around a depression in a trampoline, which is caused by the presence of a heavy bowling ball at its center. Remove the bowling ball and the trampoline springs back into shape (the depression disappears) and the tennis ball drifts away.
In this analogy, the depression represents a Gravity Well. The earth and other planets circle the sun by following the path of least resistance, so to speak (a Geodesic), around the gravity well produced by the sun at the center of the Solar System.
So, if we blow this imagery up to galactic scales, we must know that the stars and spiral arms of those galaxies are moving along the inside of a gravity well too, which is like an enormous invisible bowl in space. (Of course, it's not quite that simple, but we'll get into more detail later in this series)
In fact, the trampoline analogy above is taken from the well-known Rubber Sheet example, which is commonly used to illustrate the phenomenon of gravity. Some take issue with the example, first recognizing it as a somewhat effective aid in visualizing gravitation, but then criticize the fact that it does not fully and accurately portray it.
The problem being, of course, that the analogy employs gravity itself as an actor upon the tennis ball. Furthermore, it does not account for the dimension of time (hence, spacetime).
Yes, we know, ...that's why we call it an analogy.
My stance on the issue is that the analogy is a good one, despite clearly falling short of fully demonstrating the true complexities of General Relativity and gravity wells.