To understand infinity focus, you must first understand both what depth of field (DoF) is as well as what it isn't.
Regardless of the aperture of a lens, there will only be one distance that will be in focus. That is, there will only be one distance at which a point source of light will be focused to a single point on the recording medium. Point sources of light at other distances will be projected on the sensor (or film) plane as a blur circle, or circle of confusion (CoC). If this CoC is sufficiently small enough to be perceived as a point by human vision at a specific display size and distance, it is said to be within the DoF. The limits of DoF change based on aperture, focal length, and focus distance as well as the display size and viewing distance of the image. You can print two copies of the same image file and if one is displayed at twice the size of the other at the same viewing distance by a person with the same visual acuity the smaller print will appear to have more DoF than the larger one (assuming the resolution of the image file itself is not the limiting factor). There is no magical barrier at which everything on one side is in perfect focus and everything outside of that line is blurred. Rather, as the distance from the true point-of-focus increases, so does the size of the blur circle and we gradually begin to perceive that objects are not absolutely sharp.
Now that we've cleared that up, we may discuss what infinity focus is. Infinity focus is the point at which light rays that strike the lens as collimated light are rendered as points on the image sensor (or film). At least that is the purely theoretical definition. In practice, no light source can be perfectly collimated. Even the light emitted by lasers will spread over long distances. When your lens is focused at infinity it means that it is focused on things that are far enough away from your camera that the light rays coming from them are parallel to the degree that your lens' resolution limit can't differentiate them from perfectly parallel rays of light.
The best example I can think of is a star. Even though stars are huge, because they are so far away they appear to our eyes to be single points. Even when properly focused in a high powered telescope, they appear no larger than they do when viewed at much lower magnification. The difference between telescopes isn't how big stars are, it is how bright they are. If you want to take a photo of a sky full of stars, the only way to get them in sharpest focus is to focus at infinity. If you focus at the hyperfocal distance for a particular focal length and aperture, the stars might appear sharp at smaller display sizes, but as the display size increases you will be able to see that they are, in fact, blurry.
This brings us to your question regarding why infinity is a range on most lenses. The short answer is that variables, especially temperature, affect exactly where the focus position for infinity will be.
- As the different parts of a lens made of different materials expand and contract at different temperatures focus position will change slightly.
- Another variable for zoom lenses is the focal length. As you zoom in and out many of the parts inside the lens move in relation to each other. The position for infinity at one focal length may be in a slightly different place than the position at another focal length. Lenses that hold focus when zoomed are said to be parfocal. These are normally quite expensive, however there are many lower priced lenses that are effectively parfocal. This means they stay close enough to focused when changing focal lengths to appear in focus within the limits of the lens' minimum DoF and maximum resolution.
- Different wavelengths of light focus at slightly different distances. Back years ago many push-pull type zoom lenses had marks on the barrel for both visible and infrared light since the only thing you had to modify to shoot infrared was the film you loaded.
- In the Auto Focus era, some manufacturers, particularly Canon, have begun making lenses with additional travel beyond infinity so that the AF motor will not bump against a hard stop while attempting to focus on infinity. This puts less stress on the components of the AF system.
For landscape photos, you're normally better off using the hyperfocal distance. It will vary depending on focal length and aperture. You can always use an online DoF Calculator to figure it for a particular combination. Be aware most DoF calculators assume a viewing distance of about 12 inches and an 8X10 display size. If you plan on a larger display size, the hyperfocal distance will move further away from the camera.