The f-stop series
Lens apertures follow a √2 (≈ 1.414) geometric series, because aperture is the ratio of focal length to entrance-pupil diameter, and light passing through the lens scales with area rather than diameter. Multiplying the f-number by √2 halves the light; dividing by √2 doubles it. The standard full-stop series is f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32.
Stops between any two f-numbers
The number of stops between two f-numbers f1 and f2 is log2((f2 / f1)2). So f/2.8 to f/5.6 is exactly 2 stops darker, while f/4 to f/5 is about 0.64 stops darker.
Modern third-stop scale
Most digital cameras let you adjust aperture in 1/3-stop increments. Between f/2.8 and f/4 there are also f/3.2 and f/3.5; between f/4 and f/5.6 there are f/4.5 and f/5. These extra values are not on the chart above but appear in your camera viewfinder.
FAQ
Why do bigger numbers mean smaller apertures? Because f-number is a denominator: f/2 means the aperture is half the focal length, which is wider than f/8 (one-eighth).
Does aperture affect exposure equally on every camera? Yes — one stop is one stop, regardless of sensor size. (Crop factor changes depth of field, not exposure.)