Tearing is an graphics artifact that happens when Vsync is off. Most noticeable when you strafe, tearing will show part of the present frame and part of the last frame together, showing a shift in image.
V-sync synchronizes the monitor refresh rate and the frame-buffer in the video card so that the video card doesn't overwrite it causing tearing. The graphics card draws a frame to buffer 1 and then draws another to buffer 2 and then is supposed to come back to buffer 1 to draw the third frame, but if Vsync is on and buffer 1 wasn't emptied to the monitor yet, the graphics card will wait till that happens and then start drawing again.
Having Vsync on eliminates tearing which provides smoother image quality, however having Vsync off provides smoother framerates. It's a trade, but I prefer the smoother image quality, my eyes like it better.
You can eliminate the slower framerates of having Vsync on by utilizing a feature called triple buffering, which is mostly associated as an option of OpenGL games. This gives the graphics card a third buffer. It helps a lot, but it takes up a lot of graphics memory and it doesn't seem to be available for DX games.