Here's a simple way to think about it. If you're shooting something that's still, use Av mode. If you're shooting something that's moving, use Tv mode.
Essentially, there are three setting that affect every exposure. They are Aperture, Shutter Speed, and ISO. The aperture controls how much of the background is in focus. The shutter speed controls whether motion is frozen or blurred. And ISO controls how much noise is in the picture.
For a given amount of light, when you increase one setting you have to decrease one of the others in order to balance it out. The semi-automatic modes in the camera (Av/Tv/P) do this adjustment for you automatically. They let you control one of the settings to be how you want, and then the camera makes the corresponding adjustments to the other settings for you automatically. (In manual mode you have to do this compensation yourself.)
When you're shooting a still subject, you don't care about motion blur. So in that case, you select Av mode because that allows you to choose how much of the background is in focus or not. For a portrait or an artistic picture of a flower, pick a small aperture setting (e.g. 2.8 or 3.5). This will cause the background to be blurred, and the subject will stand out. For a landscape picture, pick a large aperture (e.g. 11). This will make everything in the picture (foreground and background) stay in better focus.
(Note that a small aperture setting, like 2.8 or 1.4 actually corresponds to a bigger hole for light to come through in the camera. So you'll see people who talk about large apertures when they mean small aperture numbers an vice versa. A big aperture has a small number, and a small aperture has a big number, but don't let that confuse you.)
Now, when you're shooting a moving subject, you first have to make sure that the shutter speed is fast enough to freeze the motion (or blur it to your liking). In that case, you select Tv mode. For slower movement you can set the camera in Tv to use 1/250 of a second. For faster motion you'll want 1/500 (or more).
The third variable in the equation is ISO. Setting the ISO basically determines how much leeway you have to set the speed and/or the aperture. If there's not enough light to shoot as fast a shutter speed as you want or as high an aperture as you want, then you'll have to raise the ISO. Raising the ISO will add grainy noise to your image. The type of camera you have will determine how high of an ISO you can use before the noise becomes bothersome. You want to use the lowest ISO you can get away with.
So, the simple story is that for still photos, chose Av and use the aperture setting to control how much of the picture is in focus. For action photos, chose Tv and use the shutter speed to control how frozen or blurred the action is. Raise the ISO if necessary (or use auto ISO) when the amount of light keeps you from choosing the settings you want.