Flash as the Main Source of Light
The first typical shooting situation is the most obvious one for flash usage: Ambient light is too low and the flash acts as an additional or main source of light.
Please see Freddy sitting in low electrical light with some candles in the background.
In such cases pictures taken with flash are a compromise between atmosphere and risk of blurs from camera shaking.
To shoot as long as possible without flash you should be familiar with setting the ISO sensitivity at your camera and possibly also the aperture. For the smallest possible blurs, i.e. short exposure times the aperture has to be as big as possible but your camera will do this already on its own if you use a fully automatic or programmed auto camera mode. And after that increasing the ISO sensitivity is the only chance to shorten the exposure time.
Alternatively you can try to fix the camera or put it onto a tripod but be aware that you can still get blurred pictures of people when your subjects move.
Another limitation you should be aware of is the limited range. It is shorter when using a small compact camera compared to more expensive ones and DSLR cameras and the small flashes integrated into the camera have less power than the external ones you plug on top of a camera.
It pays off to know the flash range of your camera and to be able to estimate if it is strong enough or not beforehand. Especially with high ISO sensitivities you can extend the flash range significantly (four times the ISO sensitivity of e.g. ISO 800 instead of ISO 200 = double flash range).