Either way they are designed to give you a picture that is correctly exposed. But you may want to control the aperture or the shutter. As an example, if someone is running and you want there to be no blur in their movement, then using a faster shutter speed will freeze their movement. It will adjust the aperture accordingly, which will allow more light in to balance the shot. The effect of this means the depth of field becomes shallower.
Av is Aperture value, or aperture priority, where you set the required aperture, and the metering in the camera will adjust the shutter speed to correctly expose the image for you.
Tv is Time value, or shutter priority, where you set the required shutter speed, and the metering in the camera will adjust the aperture to correctly expose the image for you.
P mode is Program mode I think, which will set shutter and aperture to a proper value, but should allow you to change either of them to get the same exposure, although I've not particularly used this feature.
Depth of Field, Aperture and Shutter speed are likely to be covered in numerous other questions on here, but for a picture to be correctly exposed, if you make a shutter speed one 'stop' faster, then you counteract that by making the aperture one 'stop' wider (a lower f-number), which results in the same amount of light getting in overall. The difference in the image occurs where the camera can't keep things in focus if they're at different distances away if the the aperture is wider (smaller number).
Brief example of this:
If the aperture is f22, and you focus on an object 5 meters away, objects that are 10 meters away and objects that are 2 meters away may also be in focus.
Conversely, if the aperture is f1.8 and you focus on an object 5 meters away. Objects that are 2 meters away and objects that are 10 meters away are likely to be out of focus (note these are estimates only for the sake of explanation).