If drainage from one sink comes back up in another, the two sinks are connected to the same branch line before reaching the main drain. Water will always try to remain level. If there is a blockage after the connection in the branch line, the water will automatically come back up in a sink at the same or lower level.
Grease, soap residue, and lint can build up inside your sink lines. When this buildup gets thick enough, it reduces the diameter of the pipes and causes the water to drain slowly. A hand snake is designed to remove minor clogs in traps and lines. If the sludge is especially thick, it can simply fill back in like gelatin as you pull back on the snake. If you have a major clog under the floor, you should call a professional plumber.
When used regularly, commercially available enzyme-type drain cleaners can prevent sludge from building up inside your pipes. These cleaners contain bacteria or enzymes that feed on the organic material in your pipes. Enzymatic cleaners are environmentally friendly since they do not contain any chemicals that can affect the soil or water. The downside is that these cleaners can take longer to work than chemical cleaners, are not as readily available, and have a limited shelf life.