Light stays on without switch connected
I'm going to answer based on the assumption that this is an plain incandescent and a plain old switch.
Sounds like a couple of things are combining to cause trouble. First, The switch is wired on the neutral rather than the hot side, so there's voltage at the light fixture.  Next, the return line from the light is shorted weakly to ground somewhere, thus completing a circuit and allowing the bulb to light up.  Finally, when you flip the switch, there's a better conduction circuit than the ground short, so the full voltage is seen at the bulb, and it gets brighter.

I would start by carefully removing the bulb and using a multimeter to test both the center contact and the threaded shell of the fixture against a ground point to see if there's voltage there (there should not be).