Don't take it personally, but I was always amused by how many people are impressed by the "x20 zoom" stickers and the like. IMO, this is the ultimate triumph of marketeers over the average consumer.
Manufacturing an x20 zoom costs no more than manufacturing a fixed focal length lens if you ignore parameters like optical quality, maximum aperture etc. true, it has some value to some people, but this is how marketeers divert buyers attention from parameters that increase manufacturing costs, to parameters that cost them nothing to make.
Sorry again, got carried away, I just could not resist it :er:
To your question, x8 for example, refers to the ratio between the zoom lens narrow (a.k.a. tele) focal length to its wide focal length. So, for example, Nikon's pretty good and extremely popular DSLR
18-200mm DX zoom lens would be an "x11" zoom in the p&s world, while their excellent and slightly less popular 12-24mm DX zoom lens is an "x2" zoom only. The fact that for these specific two lenses, the "x2" zoom selling price is 50% higher than that of the "x11" zoom (and the "x2" zoom does not even have vibration reduction :thumbdown

, should tell you that there is more to define a
lens, or a lens "goodness" than just zoom ratio.
Always look for the "35mm equivalent" focal length to understand what viewing angles lenses actually give. DSLRs have "focal length multiplier" numbers, typically 1.5 for Nikon and 1.6 for Canon, so, Nikon's DSLR 18mm is equivalient to 27mm focal length in the 35mm (film SLR) world, which is moderately wide, and their DSLR 200mm is equivalent to 300mm in the film world.
In the point and shoot world there are no standard focal length multipliers, the multipliers are generally higher than those of DSLRs, but vary significantly from camera to camera, even for same manufacturer, effectively, focal length multiplier numbers are only used in the DSLR world, not in the digital P&S world.
Cheers!