Aperture measurement, or "F-stop", is dimensionless (i.e. you don't express it in millimeters) and it a relative expression of the amount of light that passes through the lens at a focal length. So while it's technically read f/N, the 'N' is really the only thing you need to be concerned with as it is meant to serve as a relative constant across all focal lengths. In other words, f/6.3 represents the same relative amount of light reaching the sensor (or film) regardless of focal length or actual aperture size (i.e. f6.3 at 270mm is wide open, at 18mm it is slightly smaller) allowing for consistent exposures. It's a relative measure of light in vs. light out, so a f/1 would mean that essentially all of the light going into the front element came out the rear element at maximum aperture. The size of the opening at the end of the lens does stay the same, it's just the amount of light that reaches that opening is reduced as you zoom.
So with your 18-270mm f/3.5-6.3, what these numbers represent are the minimum available aperture at the endpoints of the zoom. At 270mm your widest available aperture is 6.3, while at 18mm it is 3.5. If you were to put the camera in Aperture Priority mode, zoom all the way out to 18mm and set the Aperture value to f3.5 and then slowly zoom in you would see the Aperture value change automatically throughout the zoom range, raising eventually to f6.3 when zoomed all the way in to 270mm. This is because as the lens elements move within the body of the lens, light is lost as a part of the shaping process, so while the elements are more compressed (as they are at 18mm) less light is lost, while at maximum expansion (i.e. when you zoom the lens generally gets longer, moving elements apart from each other) light is lost in the re-shaping process - think of it as the light "missing" the element and being absorbed by the lens wall between elements.
It is possible to have a fixed aperture throughout the focal length, like the 70-200mm f/2.8, but you will notice that these lenses have a lot more glass in them - more elements and bigger elements, which preserve the light throughout the zoom range. You'll also notice that most fixed aperture zooms also tend to be fixed length, with all movements occurring between the front and rear elements and not with the barrel getting longer.