Each mode (P, Av, Tv, M) has its own method of controlling exposure, and use. Av is good when you know you want to control the DOF with the aperture, and you don't care about the shutter speed. So under low-light conditions, the camera will keep the shutter open as long as necessary to get what it is programmed to think as the "correct" exposure. This is not what you want if the light is low, and you're hand-holding. So you have three choices: add more light (flash, move to brighter area), use the existing light more effectively (a faster lens will bring more photons to the sensor), or boost the sensitivity of the sensor (increase the ISO). Each method has its own set of characteristics. Direct flash generally makes for ugly people pictures, so "most" photographers use either bounce or modified flash to get a better quality of image. Fast lenses are great, but they do this at the expense of a very thin DOF, so focus accuracy and focus placement become very important. Increasing the ISO may cause increase in apparent noise (less of a factor on new cameras, very much an issue with older sensors).
Tv is good when you know you don't want the shutter speed to go too low, and the camera tries to compensate by varying the aperture. However, if you run out of aperture (say your max aperture is f/4), then the images will still be dark. Unless you start boosting the ISO.
P is actually a pretty good mode IF you allow the camera to control the ISO as well, and then it will pick whatever combination of shutter-speed, aperture, and ISO that gives a "correct" exposure. However, with this method, you've handed control of most of the decisions to the programming in the camera, and it may or may not actually work for your specific situation.
M is of course, where you set everything yourself. This is a good mode to use when you know what you are doing, and can predict the effects each of your exposure choices (shutter speed, aperture, ISO) will have on the final image.
Keep in mind that the camera needs a minimum amount of light to be able to carry out its AF functions, and if the light is too low, the camera may be incapable of getting a focus lock, unless it gets additional assistance.