ASA is the old name for ISO.
It depends if you're using film or digital photography. If it's film, then the ISO number is a rating telling you how sensitive the film is to light. The higher the number, the more sensitive it is, and the less time needed to expose a photo. So, it's best to choose a higher ISO for indoor or night photos, or wherever there is typically less light - however, high ISO films tend to produce a grainier effect on your photos.
If you're using digital, then this is the camera simulating the effect of different ISO films.
Film speed standards.
ASA = American Standards Association
There were also separate British, German and Russian sets of standards.
(British Standards Institute, Deutsches Institut fur Normung, and the GOST of Russia)
ISO = International Standards association and they combined the American and German scales for the ISO standard numbers.
The American scale ASA was based on a direct mathematical scale while the German system gave film speed on a logarithmic math scale.
ISO gives both scales.
You have the ASA style numbers from 6 to 102,400.
You have DIN style numbers from 9° to 51°.
Note that ASA numbers above 12,800 or 42°DIN are only on digital cameras because film is not able to give the same sensitivity.