Filters other than the Cokin type come in a specific diameter (49mm, 77mm, etc) to fit onto the front of your lens or lenses. Check the lens to see what diameter you need. If you have more than one lens and they have different diameters you can either buy one of each size, the larger size only with a step-up adapter ring (or is it step-down?), or go with the Cokin system and just get the mounting rings for the sizes you need.

Nothing wrong with your camera; it's a limitation of photography. Any camera will overexpose if you shoot in bright sun for one second and can't stop down far enough for proper exposure.

ISO + shutter speed + aperture = exposure. It's an equation - change one value, and to keep the same results you'll have to change another value to compensate for it. For example, if you're at ISO200, f16 and 1/250 which should be pretty close to mid-day sun and you change the shutter speed to one second without changing any of the other settings to compensate for that long exposure, you'll give the sensor so much light that it can't record anything. Going from 1/250 to one second is adding eight stops of light. Even negative film which is more tolerant of overexposure than digital can't handle anywhere near that. Maybe three stops if you're lucky but with digital to really be right you need to be within one stop. That's the idea behind the neutral density filter - it's another way of blocking how much light gets to the digital sensor.