Tearing is when your graphic card calculates a new frame while drawing the old one and mid way through drawing the old one continues drawing with the new one, it will give you one (or two) BIG jumps on any object running through the tear, like if you tear a paper and put it together moving one part 3 cm.
This does not happen when vsync is on, since vsync synchronizes the frame changes with the redraw on the monitor, so you dont get part old frame part new frame.
Anti Alialising will transfer the pixels along a angled line with high contrast into a series of pixels with intermediate colors, by that tricking the eye so it looks smooth. Thats what you want, but it costs some power on the graphic board so if you switch on 8x AA your frame rate probably goes down noticeable.