Everyone knows what gravity is, things that have mass are pulled due to gravity. But that is not the only thing gravity does. It does many other things. It causes any mass to distort space around it by a minuscule amount. So in daily life you are never gonna see it. But when something is really massive , like the sun, or a black-hole or an entire galaxy, this effect is very pronounced.
This curved space-time can act like a lens. It can magnify and distort the images of the objects behind it. So, anything with a huge mass is a potential telescope for things which lie behind it. This effect is called Gravitational Lensing. Not only can it magnify images, it can also make multiple images of an object. Empirical verification of this phenomena was used as proof for the correctness of Einsteins theory of relativity as a theory that describes nature.
The more massive the galaxy the more it will bend space-time around it and the more distorted the image of the object behind it will be.
Gravitational lensing is a very useful concept and it has many implications.
If you see a displaced image of a galaxy whose real position is known from a different source. Then using that displacement you can calculate the mass of the object that caused it. Its very easy to calculate how much mass is needed to bend space-time sufficiently in order to generate a displacement that we calculated from our previous data. This is how the mass of various objects in the sky is calculated, although there are many other techniques.
But what if you measure the mass of an object using this technique and you find out that it is different than the mass that we see using other techniques. Then there is a discrepancy. This Discrepancy in the measured mass of galaxies is called Dark Matter.
When we measure the mass of a galaxy using Gravitational Lensing it turns out to be much more than we can see. And so, this is one of the biggest evidence for Dark Matter.
There are other sets of evidence for Dark Matter and one of them is the rotational velocity of galaxies.
Almost all galaxies rotate around their axis along with the stars inside it. But the stars in the galaxies appear to be moving much faster than they should. In our solar system planets closer to the sun have higher speeds and than the ones which are more far away. This is what you expect, as things more far away are less influenced by gravity. But in a galaxy, all the stars move more or less at the same speed.
There must be some more mass hidden in the galaxies which is providing enough gravitational tug to the stars and preventing them from flying into intergalactic space.
Another explanation could be that our laws of gravity are incapable of dealing with such huge distances and mass scales and need some modifications. This would mean that dark matter doesn’t really exist its just that our theories are wrong. But the scientific consensus on this in the light of many other studies and phenomena is that maybe the theories may need some modifications but Dark Matter nevertheless Exists.