Lots of different things, at lots of different points in his career.

Special relativity came about in part due to the results of the Michelson-Morley experiment. It was well-known at the time that light was a wave, because you could diffract it into a spectrum (read: rainbow) like a wave, and get two light waves form an interference pattern as they interacted with each other (like the way the ripples interact when you throw two stones into a pond). However, waves need some medium to travel through (for instance, ocean waves flow through water, and sound waves travel through air). What was the medium through which light waves traveled? Although there wasn't any direct evidence for it, it must be there somewhere, and people called it the aether.

Enter Michelson and Morley: they reasoned that the earth must be traveling through this aether. Consequently, we could expect that light would speed up if it travels through the aether in the same direction the earth travels, and would slow down when traveling the opposite way (think of the Doppler shift, and how cars traveling towards you make a higher pitch sound than cars traveling away from you). They built a device which would send light in two perpendicular directions and then very sensitively measure the change in speed between the two. They expected that the light beam pointing in the direction the earth was traveling around the sun would have a different speed than the beam pointing in the other direction. To everyone's surprise, this was not the case! Instead, the light seemed to travel at the same speed in all directions. It kinda looked like the aether didn't exist (or at least, did not act the way other wave-carrying media did)!

Einstein came up with Special Relativity in part by accepting the results of the Michelson-Morley experiment. The two premises of SR are 1) the laws of physics work the same in any inertial reference frame (i.e., it doesn't matter where you are or how fast you're going) and 2) the speed of light in a vacuum is a constant, no matter what direction the light is traveling. From those two assumptions, all the rest of Special Relativity follows.

Einstein explained various other experiments with various other results. For example, he won the Nobel Prize for explaining the photoelectric effect. It turns out that when you shine high frequency light onto a metal, it emits electrons. The puzzling thing was that if you changed the intensity of the light (by making it brighter), the electrons emitted did not change energy (though there were more of them), whereas if you changed the frequency (read: color) of the light, the energy of the electrons would change.

If light were actually a wave, this result makes no sense. Large (high-intensity) waves should have more energy and thus impart more energy onto the electrons they knock loose, the way large ocean waves during a storm are more dangerous than small ones during a calm. On the other hand, changing the frequency (how closely the waves are packed together) shouldn't obviously do anything. Einstein pointed out, however, that if light acts like a particle (call it a photon), the photoelectric effect makes perfect sense: varying the intensity changes how many photons you have, though each one has a constant amount of energy. Consequently, you'll vary the number of electrons you can knock loose, but not the energy imparted from a single photon onto a single electron. Meanwhile, varying the frequency changes how much energy each photon has, and thus how much energy the photon can give to the electron it hits.

We could also talk about General Relativity, and how it explained pre-existing mysteries like the way Mercury's orbit around the sun precessed around like a spirograph toy, flying in the face of Newtonian physics, and how GR predicted other phenomena that would soon be confirmed, like the way massive objects like stars and galaxies distort light that passes near them. but this comment is long enough already.

I'm not a historian, I have studied a lot of physics though.

There were many experiments in to the nature of light and how it was affected physical objects. Even from the early 19th century. Many of the experiments that attempted to prove the existence of the luminous aether (a substance that carried light waves like air carries sound waves) were producing results that were explained by Special Relativity.

One of the earliest was in 1810 when Francois Arago tested a hypothesis that the refraction of light from stars would be different due to their relative velocity with respect to the Earth at different times of the year.

The experiment looked at the refraction of light through a glass prism using stars as a light source. It was hypothesised that it would vary from star to star and at different times of year. (https://en.wikipedia.org/wiki/Aether_drag_hypothesis)

He found that they were all the same. This is explained by Special Relativity, but the time it was eventually explained by some sort of drag in the aether.

This Wikipedia article for it is quite unclear on what the experiments actually were: https://en.wikipedia.org/wiki/Tests_of_special_relativity If you're up for some googling and additional reading it's a good place to start.

Explaining the Photoelectric Effect was a big one for him. His work on that won him the Nobel prize and also substantiated Max Planck's quantization of energy.. really kicking off the quantum revolution. In fact he was very active for many decades, and he had his fingers in a lot of pies aside from relativity: Brownian Motion, disregarding the luminiferous aether concept, atomic structure, wave particle duality, full blown cosmology - he was a pretty versatile chap.

Maybe /r/askscience would be better equipped for this