Alright. First let us begin with dark matter in this post, which in many ways is the simpler issue. There is all kinds of evidence that all the matter we see in the universe is not all there is. What is some of this “evidence”? Let me try and give you a couple of examples. One of them is associated with “galactic gravitational potentials”. What does that mean? Now, suppose I was observing a galaxy in my telescope. I saw a star that was far from the central bright core of the galaxy. Then, I expect that the star will have a velocity (GM/R)^1/2, where R the distance of the star from the center and M is the mass of the bright stuff in the middle . Now I measure the velocity of this star, it is moving much faster than this estimate. You might say “Aha! You are just underestimating the mass of the bright object in the middle!” But, if you believe that the universe is homogeneous (same everywhere) and isotropic (same in every direction you look), you have no choice but to conclude that there is just some universal parameter that you have to fit observed data, called the “Mass to light” ratio (and you have no choice but to believe this hypothesis unless you want to also believe that the earth is the center of the universe somehow). And I urge you to go and play with this applet to see that you CANNOT fit the observed curve with just one such parameter. So, there must be something else. What that something else could be is some mass that I cannot see, that I don’t know anything about so far, such that the star that I think is far away from most of the mass in the galaxy is not so far at all, for this stuff I cannot see is filling the intermediate space that appears to me to be empty. So, the postulation of the existence of this “dark matter” is one possible explanation for these weird velocities of apparently far-flung stars in galaxies.
One more piece of evidence is associated with the mass of clusters of galaxies. This is rather involved, but if you want, you can go read about it in this post on Cosmic Variance . So, let me move on to another piece of evidence. This is associated with large scale structure in the universe (this is just jargon for stars, galaxies, you and me). The way this argument works is as follows. We know how the universe is today. We use our telescopes, optical or otherwise and know the mass density in the universe everywhere. We also know what the mass density in the universe was when it was only 400000 years old (that is very young on cosmological time scales). This info comes from the cosmic microwave background . Then knowing the mass in the universe, and knowing the laws of gravity, I should be able to go from the scenario 400000 years ago to now. But, I cannot. It turns out that if I try to so this, I get a mass inhomogeneity much smaller than what we have today, to the extent that you and I cannot be here. But we are here. So, one possible explanation could be that there is this “dark matter” we invented earlier is there in the early universe and the information about its distribution is not in the cosmic microwave background and hence we are not able to get to the present structure of the universe and the fact that you and me are here.
Do you see? The postulation of this “dark matter” solves many problems that are around in astrophysics and cosmology. But, the problem is that we don’t know yet what this “dark matter” made of and how it talks to the regular matter that you and I are made of. We have ideas as theorists and we have experimentalist out there testing to see if any of these ideas hold water. But until then, we just have to live with “dark matter”! 
 There are a whole bunch of erudite articles on the web, for example, this one by Sean Carroll. I will try to be very minimal here, no way near as erudite.
 You can do an itty-bitty circular motion calculation to see this, given that gravity leads to acceleration GM/(R^2) on a particle at a distance R.
 There is also an interesting comment thread here that is a back and forth on dark matter, that we (readers and writers) at scientific curiosity will do well to emulate! :)
 This cosmic microwave background comes from an event in the past of our universe that is called decoupling. But I have to do a lot more work to get this point across. I will provide an update with some appropriate reference subsequently.
 But this, namely the postulation of the existence of dark matter is not the only way out of the many problems that cosmologists face. But the other ways out are deferred to a subsequent post coming up shortly, so stay tuned.