In a lens description, the f/number refers to the maximum aperture of the lens. I won't go into detail on apertures themselves, as its answered elsewhere, so make yourself familiar with that term before reading this.

If the lens has a single number, e.g. "17-55mm f/2.8", it means the maximum aperture for any focal length along the zoom range is f/2.8. The maximum aperture at 17mm and 55mm will be f/2.8.

If the lens has a range, e.g. "18-270mm f/3.5-6.3", it means the maximum aperture varies depending on what focal length you're using. On the lens you describe, the maximum aperture is f/3.5 at 18mm, and f/6.3 at 270mm. As you zoom in (from the wide 18mm to telephoto 270mm) the maximum aperture will decrease. At 35mm it might be f/4, at 100mm it might be f/5.6. The specifications rarely indicate how the aperture varies, but it's almost certainly not linear.

Some review sites will list how the maximum aperture varies across the lens range, for example this review of the Canon 18-200mm f/3.5-5.6 includes a table showing the maximum aperture is:

• f/3.5 from 18mm to 23mm
• f/4.0 from 24mm to 39mm
• f/4.5 from 40mm to 49mm
• f/5.0 from 50mm to 89mm
• f/5.6 from 90mm to 200mm (note this is a big chunk of the zoom range)

Likewise, the minimum aperture (largest numerical value) will vary in line with the maximum aperture (smallest numerical value), however that's generally considered less important (e.g. minimum aperture might go from f/22 to f/36 over the same range as the maximum aperture varies from f/3.5 to f/5.6).

Whether a lens has a constant or variable maximum aperture depends on the design. Typically, the constant maximum aperture lenses are larger, heavier, and more expensive than their variable maximum aperture counterparts, as having a wider aperture at the long end of a zoom lens requires a larger entrance pupil (the aperture as seen through the front of the lens), which means a physically larger aperture and/or greater magnification at the front of the lens (which usually means big front elements). The moving elements within the lens will also be subject to different restrictions in either case (e.g. aperture and front elements generally need to move further apart for a telephoto wide aperture lens).

Here's another Photo.SE answer that may give more info on constant maximum aperture lens designs.