Dark matter is a hypothesized type of matter which we can't detect, but we can see how it affects the universe around it. It is thought to be made of Weakly Interacting Massive Particle (WIMP). Just like dark energy, dark matter i.e. WIMP doesn't interact through the electromagnetic force. That means we can't see it in visible light, we can't detect it with any kind of telescope, irregardless of the frequency at which it operates, all because the electromagnetic force carriers, photons, can't give us any information about it. Dark matter is thought to interact only through gravity and weak force.
The reason we think dark matter exists, the reason this isn't just some theory that some people find true – it is currently a consensus among cosmologists that dark matter is real – is the overwhelming evidence of its influence on visible matter.
It was first theorised in 1930s when it became obvious that the mass of visible matter is not enough to account for the way stars in our galaxy move and how galaxies move when grouped together in clusters. Even the way galaxies rotate indicate they are heavier than they look.
This isn't just a minor difference in mass. Current calculations put mass of dark matter at 26.8% of the mass of whole universe. Visible matter only accounts for 4.9%.
For example, if visible matter were all there is in the Milky Way galaxy, outer parts of its spiral arms would have lost their structure. Take any kind of disk for example. A DVD will do. If you rotate it, a point at its outer part has to move faster to catch up with another point near the centre in order to simultaneously complete one rotation. A rotating, spiral galaxy doesn't have the same solid build like our DVD though. Galaxies are made of stars with mostly empty space between them. If you rotated a galaxy that only has visible matter, its outer parts wouldn't be able to keep up with inner parts. The farther away the point is, in this case a star, the slower its radial velocity should be. This isn't case with galaxies, so there has to be extra matter that gives galaxies that much needed robustness, meaning that although it isn't as solid as the DVD in our example, it still has more to its structure than we can observe. These observations indicate that stars in outer parts of spiral galaxies move at the similar speed to stars in inner parts (but not faster as is the case with our DVD).
Second evidence comes from the way galaxies group together in clusters. Their movement speed suggests they are bound together by more mass than visible matter can possibly have. Alternately, gravity is stronger force than we think, but the calculations are based on the theory of general relativity that is proven to work over and over again. As far as we know, laws of physics are the same in all parts of the universe.
Another evidence is offered by a phenomenon called gravitational lensing. It is the way a massive object in the foreground distorts the picture of another, more distant object. We know that massive objects bend space, so when the light passes through this space, it changes its direction. This way we can see objects behind the massive object in the foreground, although we don't have a direct line of sight to these objects in the background. A cluster of galaxy is a typical example of such gravitational lens. Observations have shown, however, that these clusters bend light i.e. space more than they should if they only had mass originating from their visible matter.
Although cosmologists have no direct observational evidence, they think that a subatomic particle named Weakly Interacting Massive Particle (WIMP) is what dark matter is made of. WIMPs are very elusive, thought to interact with visible matter only from time to time, if they do at all, and right now scientists are trying to find evidence of this interaction or an interaction between two WIMPs. In many way, WIMPs are similar to neutrinos, but far more massive and thus slower.