Many people made movie-camera like machines in the 1890s, but Edison is generally credited as the inventor of the first commercially viable movie camera. He was given the idea by Muybridge and the film by Eastman.
Actually, it was his assistant W.K.L. Dickson who did the work.
Who "really" invented movies? Muybridge himself had invented a way of projecting his serial photographs, a machine called the zoopraxiscope, and used it as early as 1880. But, Muybridge was still just projecting serial photographs, not real movies. One of the most interesting claims of early motion pictures is that of Louis Le Prince. Le Prince was born in France. His father was a friend of Jacques Daguerre and Le Prince grew up interested in photography. He moved to England and established the Leeds Technical School of Art, where he taught photography. He is credited with several inventions including a 16 lens camera that took 16 images in sequence on a single glass plate. Le Prince also built and used a basic single lens movie camera. He produced what is probably the earliest thing that we could really call a movie in 1888. You can see the entire 2 seconds of it by clicking here. The camera worked using paper backed film. Le Prince also created a projector to show his film. He was scheduled to do a public demonstration of his work in 1890. But, it never happened. Le Prince went to visit his brother in France. In September 1890, he got on a train from Dijon to Paris and was never seen again. If you love conspiracy theories, the disappearance of Le Prince is a treasure trove of them. Some people think he committed suicide, some that his brother killed him in a argument over a will, some that Edison had him killed, some that he was murdered by robbers, some that he was gay and disappeared for a new life. Whatever is the case, he hasn't turned up in the last 130 years or so.
Here are my thoughts on the controversies about the invention of movies. I'm not a fan of Edison, but ultimately he gets the real credit. All of his predecessors, including Le Prince produced partial devices that, if everything had fallen in place, might have led to movies. Edison, whatever his faults, had the vision to bring all the different elements together to create a replicable, commercially viable product. By comparison, others, including the Lumieres (who we'll talk about next) had very limited vision. And, the notion that Edison had Le Prince killed is just plain silly for two reasons. First, in 1890, no one had any idea whether or not movies were important or valuable. They were just one cool idea in a sea of cool ideas. It's only retrospectively that we can see them as a critical invention. Second, movies were just one of many, many things Edison was working on, and far from the most important. If Edison had dispatched killers to take out his competitors, the world would be strewn with Edison related corpses and that's not the case.