The shutter speed (1/60, 1/200, etc.) controls the time the shutter is open thereby controlling the amount of light let in. If you were shooting fast action such as sports and you wanted to freeze the action, you would choose a high shutter speed (ex. 1/1000).
You can make the picture darker by changing the f-stop, but that will also change your depth of field (DOF). If you wanted the subject to be sharp and the background blurred out, choose a wide aperture (ex. f/4.0). The wider (lower f number) the aperture, the more blurred the background will be. Try it on your 60D. Take a still shot of something on a table with something else recognizable that is sitting across the room. Take a photo using the widest aperture and then the narrowest (ex. f/22) and see the difference.
If you're in a dark situation to start, such as a nighttime or a dark room most people open up the aperture as wide as possible (low f-number) to let in as much light as possible while still keeping a suitable shutter speed that they can hand hold the camera.
Sports shooters often use a wide aperture combined with a fast shutter speed to make the subject stand out while still freezing the action. There are a ton more examples of when to use f-stop or shutter speed to control the light, but those are the easiest to understand.