The reason why people think that there is dark matter is very simple actually. You are probably aware that the stars in a galaxy revolve around the centre of that galaxy. Based on what we know about physics, we would expect that the stars far away from the galactic center move slower than the stars closer to the galactic center, as illustrated in the picture by the lower curve. However, our observations show that the reality follows the upper curve. Now if we assume that there is such a thing as dark matter, and we give it a certain distribution, then we can predict that the upper curve is the correct one, using what we already know about gravitation and stuff like that.
It is difficult (for me) to say how far we are from proving the existence of dark matter. There are numerous experiments running, but none of them have found conclusive results, merely encouraging results.
We don't know what dark matter is made of (if it even exists), but there are a number of candidates: WIMPs (weakly-interacting massive particles), neutrinos, and axions, and probably some more. What we do know is that, if it exists, it interacts only very little with "ordinary" matter.
If you are interested in reading more about dark matter and dark energy, I would advice you to search for the NASA website, this page https://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy
might be a good starting point. If you have access to a university, I would recommend you to borrow or download (but only if you can get it for free) the book Introduction to Particle and Astroparticle Physics, by De Angelis and Pimenta. It discusses dark matter and a lot of other topics.