The video game, Tom and Jerry: The Movie was a tie-in to the movie of the same name in 1992. The game was released for the Sega Master System and Game Gear and appears to have no real relation to the movie other than the name and starring characters.
Like most licensed games, it is really mediocre at best. You get to play the role of Tom as he chases Jerry down in a series of levels. If you are particularly good, you can catch Jerry and end the level early. Otherwise, you have a puzzle to solve at the end of each level which determines whether you catch him or not. There are a variety of environmental obstacles to avoid or overcome and Jerry will of course make your job harder. If Jerry gets out of sight then you lose.
While the game play of the Sega Master System and Game Gear versions are almost identical, the levels are completely different. If you like one, you will probably like the other. Having said that, unless you are a big Tom and Jerry fan, then you should probably avoid this one.
As far as I know, this game has not been re-released so you'll have to track down an original or use emulation if you want to give it a try. As far as getting an original, the big screen (SMS) version is far preferable to the Game Gear version.