I don't know if this is the best sub; r/askscientists might give you a more complete answer but I'll try and give the best explanation I can.

First of all he is one of the most revolutionary physicist of the world so I'll give you a brief run down of his most important achievements. It all started in 1905, the so called annus mirabilis, when he, at the time an employee of the patent office of Bern, wrote in one year four incredibly remarkable papers:

• the first one regarded the photoeletric effect, which the emission of eletrons when radiations hits a surface. In this paper he used a theory proposed by Plank a few years earlier to explain this effect, that is that energy moves in "packets" of energy (which we now call photons) rather than being just a wave as people thought at the time.
• the second regarded the brownian motion, that is the motion of small particles (like pollen) in suspension in a liquid. About 70 years earlier, a british botanist had observed that pollen in suspension moved all around for no apparent reason, einstain proved that the movement of the atoms caused by heat could explain that phenomenon, thus proving Boltzmann theories on thermodynamics and proved once and for all the existence of atoms.
• the paper on special relativity where he changed physics by introducing the idea of spacetime and that nothing can be faster than light and that light (in a vaccum) goes to the same speed for any observers. From that he showed that a lot of stuff really cool (like that time can dilate and simultaneity is relative) and solved the contrast between newton mechanics and maxwell theory of eletromagnetism. Moreover with his theory he managed to explain that there was no need for a medium for light to travel (which scientists at the time where still looking for after a failed experiment to prove its existence, the michelson and morley experiment).
• finally as a conequence of the previous he prove the mass energy equivalence ( e=mc2 ) which basically says that a particle with mass possess energy, called "rest energy" distinct from kinetic and potential energy. This implied that if a particle lost mass, it would also have to release energy which is the basis for the nuclear reactions.

and all this in 1 year and all this discoveries gave him a lot of fame.

Later in 1915 he wrote about general relativity, where he explained gravity (which was notably left out by special relativty) as a curvature of spacetime (it is called special relativty because it regards the special case of flat space time). While special relatively was quickly adopted by the scientific community, general relativity was controversial for a while but now is the main theory we have about garvity, time and space.

He was also crticical in the development of quantuum physics (notably he had a long debate with Bohr, one of the fathers of quantum physics during the Solvay Conference where each day he presented problems about Bohr theory and each day Bohr solved them) but he abandoned the field because he did not believe that we could only have probabilistic guesses on the behavior of particles rather than deterministic rules (which instead at the moment is the main theory).

He also wrote other stuff, but I don't want to dilungate so I just presented the main results.

So why is he so famous? First of all because of all those discoveries, in a few years he outdid in discoveries what a faculty of physics probably will do a century. This is not to say that he did not make mistakes, notably when he noted that the universe was expanding according to his theory he added a constant in the formulas to keep it still (now we know that the universe is in fact expanding), he also refused to accept the conclusions of quantum physics while,as far as we know, they are valid, moreover in his later years he conducted a long and unfructuful research into a unified theory of the universe.

He wasn't alone either, as I have shown most of his discoveries where based on the work of others such as Maxwell, Plank and especially Poincarè, a mathematician on whose work his theory of relativity is based upon. Still he was probably the brightest mind of the century, but he also had extremely good PR.

He was a nice and bright person, he liked the violin, he was interested in politics and championed several causes he liked such as the fight against racism, he had a wholesome view of the universe and a inspiring personal philosopy and thanks to his extravagant look (with the messy hair and mustaches) he perfectly impersonated the stereotype of the distracted genius. All those factors made it so that he became extremely popular in the media to the point that his name is now a byword for intelligent.