When my mom asked why it took me so long to compose a shot, I told her because I had to take a lot of things into consideration like ISO speed, F-stop, and shutter speed. She said “I don’t know from F-stops.” Well, here it is.
The short answer is that an f-stop is a discrete point along the continuum of aperture diameter in a photographic lens. So then what’s an aperture and why does its diameter matter to photography? That’s a MUCH better question!
The aperture in a lens is a dark opaque element with a variable-sized hole in the center. It’s composed of several (5-9 usually) overlapping blades that, when adjusted, change the diameter of the center hole. The aperture opens and closes to allow more or less light in as desired. In addition to varying the amount of light, it restricts the light’s angle of incidence when hitting the sensor. This is called beam collimation and directly affects the depth of field (how much of the shot is in focus). Think this way: If you have a bundle of spaghetti and hold it loosely (wide aperture), the strands of pasta point all different directions. If you hold it tightly (narrow aperture), the strands all point in the same direction. Light beams all pointing in the same direction gives a broad depth of field with almost the whole shot in focus. Light beams pointing all different directions means a narrow depth of field with only a tiny bit of the shot in focus and the rest blurry.
So that still doesn’t answer what an f-stop is. The f-stop was originally one of the “stops” on the aperture diameter control ring on a manual lens. The number corresponding to the stop is given thus: Φ = ƒ/ N, where N is the f-number, ƒ is the focal length of the lens and Φ is the diameter of the aperture. F numbers can be given either as a fraction (f/N) or as a ratio (1:N). Either way, the f-number is just the denominator in the focal equation. Given that, it then makes sense that the lower the f-number, the wider the aperture (and the narrower the depth of field). So lets take an example: A 50mm lens has a maximum aperture size of f/1.4 and a minimum aperture size of f/22. f, in this case, is 50mm. The widest the aperture will open in this lens is therefore 50mm / 1.4 = 35.71mm. The narrowest it will close to is 2.27mm. The lens’ sweet spot happens to be between f/8 (6.25mm) and f/11 (4.55mm).
So the bottom line is than an f-stop is a setting on the lens that controls how wide the aperture opens. A wider aperture means more light but a narrower depth of field and a narrower aperture means less light and a wider depth of field.
See the Wikipedia article for another (probably better) explanation.