The 6x, 8x,10x etc is just a selling point used to market compact cameras to people who know nothing about photography, but believe in the hype of the bigger the number, the better. The numbers only refer to a comparison of the focal length of the lens at either end of it's ability. Without knowing what the focal length is at either the wide end or telephoto end, the 6x or whatever is pretty meaningless.
In DSLR terms, for example, you could have a 20-100mm zoom lens, and a 100-500mm zoom lens. In compact parlance, both are 5x but other than that, they would be like chalk and cheese.
In old 35mm film days, a 50mm lens was considered standard, and was supplied with most new cameras. In simplistic terms anything shorter than that was considered wide angle and anything longer was considered telephoto. If you were to take several photos from the same viewpoint, a wide angle lens would allow you to capture more of the scene, as it has a wider field of view. Conversely, a telephoto lens would capture less, allowing you to hone in on a specific part of the scene, a bit like looking through a pair of binoculars.
If we accept a 50mm lens as being a standard of some sort, it makes it easier for us to imagine what lenses of other focal lengths will give us; a 500mm would be 10x whereas a 24mm would be roughly 1/2x. I know purists will probably frown on this notion, but for a novice who has no concept of what different focal lengths expressed in mm actually mean it's as good a place to start as any.