Clunking on 1995 Chevrolet Cavalier

When I put my car in gear the passenger side clunks loudly and when someone is sitting there they say that they can feel the floor move on that side. What causes this and how do I fix it?

Asked by for the 1995 Chevrolet Cavalier
This sounds like it could be a defective engine mount or defective transmission mount.
So does that mean that I need to take it in to a mechanic to be fixed? or is it something that won't really bother the car to much?