What is dark matter and how scientists understood it's existence?
Short answer? We have absolutely no idea, and galaxies spin (way) too fast for their visible matter to hold them together.
We might be better dealing with these the other way round - firstly it was noticed, shortly after we discovered that the many ‘clouds’ (nebulae) we noticed in the night sky were actually galaxies, that they were spinning. This was discovered by using the Doppler effect on spectroscopic images of galaxies, which showed one side of a galaxy approaching us and the opposite side receding.
So far, so happy, they spin. Then Fritz Zwicky, whilst examining the Coma cluster of galaxies in 1933, spotted that the galaxies were spinning too fast for the visible matter to generate sufficient force to hold them together. This was confirmed in the 1970’s by Vera Rubin Cooper, specifically, the outer rim of the galaxies would be expected to rotate much slower than the centre according to Newtonian mechanics. This was not observed, the rotational curves were almost “flat” in the outer reaches of galaxies.
This allowed for two possibilities: (a) Newton’s theory of universal gravitation was wrong (b) there existed vast quantities of ‘dark’ matter. Nobody seriously believed (a) so we were left with the hypothesis that a dark matter “halo” surrounded each galaxy.
Now for the second part (your first) we have searched for the particles that might make up dark matter very hard (a Nobel prize almost certainly awaits the discoverers) but despite many year stars of searching (both in space and in particle physics labs) nothing has been found that could possible fit the bill.
Neutrinos of various sorts have been proposed, as have exotic matter, stable quarks groupings, dead stars, black holes etc etc. Martin Rees (Astronomer Royal at the time) even went so far as to suggest that it could be unread copies of the Astrophysical Journal! We genuinely have no idea and it bugs us. A lot.
It appears that this stuff outweighs visible matter by a factor of about 5:1 across the universe and you have as much idea what it is as they do. In my view this makes it an excellent time to be studying either cosmology or particle physics, because I hope major discoveries are just around the corner.