The camera will calculate the amount of flash to use in one of two ways, depending on the conditions.

The first case is when the flash is the primary light source. In this case, the shutter speed is irrelevant. The target exposure is still the standard 18% gray, but flash power effectively replaces shutter duration in the calculation. That's because the ambient light is low enough that any moderate shutter speed just wont let in enough to make a difference: only the brief powerful flash impulse will matter. The camera's program calculates the ISO, aperture, and flash power needed. (In some modes, you may supply ISO or aperture and the camera will compute the rest.) In modern TTL systems, the amount of flash power needed is measured with a very brief, lower-power "pre-flash" (possibly combined with the focus distance info from the lens). If you're calculating it yourself, you use the guide number for the flash (adjusted for ISO) divided by distance to the subject to find the aperture to use.

The second case is when the flash is used for fill. Here, the ambient light is bright enough, sensitivity is high enough, and aperture wide enough that the shutter speed actually does have an impact on the exposure. The way to set the exact balance between shutter speed (ambient light) and flash power (your additional fill) will vary between camera models and camera modes, but that basic principle is that there's some balance between light sources. The scene without flash would be slightly underexposed (but not completely dark), but the flash burst uses a power level intended to bring it back up to "correct" middle exposure. If you're making the calculation yourself, basically you calculate as above but then subtract a stop (or half, or two, as desired), compensating with the ambient exposure. (More on that under How do I manually calculate fill flash?)

When exactly the camera uses one approach over the other is also brand/model specific; usually putting the camera into aperture- or shutter-priority mode makes the camera expect you want fill flash. It may also make that decision automatically in other automatic modes when the ambient light is high.