The main lines of evidence for dark matter:
* Galactic rotation curves
* Velocity distribution in clusters of galaxies
* Gravitational lensing in general
* The Bullet Cluster in particular
* The pattern of positions of galaxies in the universe
* The pattern of Baryon-acoustic oscillations in the cosmic microwave background and in the galaxy distribution
* The primordial distribution of light elements in the universe
We know of some kinds of dark matter already: There is a huge amount of neutrinos left over from the big bang, and since these interact very weakly with other stuff, they definitely qualify as dark. Other known kinds of dark matter are black holes, and compact, cold objects made out of baryons (normal matter). So dark matter exists.
The problem is that there isn't enough of the normal kinds of dark matter. To match the pattern in the cosmic microwave background and the amount of hydrogen, helium and lithium in the universe, one needs by far most of the dark matter to be non-baryonic (i.e. not normal matter, but something like neutrinos, but heavier). This kind of dark matter is something we have to postulate exists in order to match observations. But when we do assume it exists, the theory matches observations extremely well. As an example, look at the CMB power spectrum as mesured by Planck. The error bars are so small that you mostly can't see them, and the points lie smack on top of the theory curve. But only if dark matter is included.
And it just so happens that the amount of dark matter that makes theory match the points in that graph also makes the element abundances, galaxy distribution, lensing observations and galaxy cluster velocities work too. Such a coincidence is pretty telling, I think.
But yes, people have tried to avoid dark matter by modifying gravity instead (though nowadays, the most common motivation for modifying graivty is to avoid dark energy). MOND is an example of that. MOND is like normal Newtonian gravity as long as the gravitational acceleration is large (like in the solar system), but instead of falling to arbitrarily low values as distances increase, the gravitational acceleration has an effective minimal value that it approaches as you move away. And such a constant value is just what you need to get the flat rotation curves we observe in galaxies. Which is the problem MOND was invented to solve.
MOND is an elegant solution for galaxies, but it loses all its elegance and predictive power when you try to apply it to the other areas where dark matter shows up. And in some cases it is plainly ruled out as an explanation. MOND, like Newtonian gravity, is a central force, which means that the force points towards the mass that generated it. But in the Bullet cluster, the gravitational force points towards areas with little visible matter, away from areas with much visible matter. This is impossible to fit into MOND. So the Bullet cluster basically killed MOND.
Some of MOND lives on in TeVES, which is an attempt at a relativistic version of MOND. Sadly TeVES has none of the simplicity and elegance of MOND, and while it can explain the Bullet cluster, it effectively does to by using dark matter.
So as I said: It is hard to avoid dark matter. In general there are are large numbers of theories of modified gravity, but all of them are more complicated than general relativity + dark matter, and none of them match observations any better than it.