When the drainage from one fixture backs up into another, it means that the two lines are connected on one branch line before it enters the main drain. Water always seeks its own level so it will come up in a lower fixture or another sink at the same level if the branch line connecting the fixtures has a blockage after the connection.
Sink lines get filled up with lint, grease and soap scum from the fixtures. As the water goes down the drain, a film is left on the inside of the piping, eventually building up so thick that they reduce the internal diameter of the drain and slow the water flow. A hand snake is designed to open minor clogs in small traps or branch lines. A hand snake can punch a hole in a clog, but it is often like punching a hole in gelatin. The sludge builds up so thick, when you pull the snake back the sludge just fills back in. Hand snakes are not designed to open piping under the floor, so if the clog is there, you'll need a professional plumber.
Once the line is cleaned you can use an enzyme-type drain cleaner like Roto-Rooter Pipe Shield® to eat away biological material and prevent it from sticking to the inside of the piping, which keeps the line open and flowing. Pipe Shield® is available only through Roto-Rooter.