There are two ways "infinity focus" could be taken — "an infinite amount of focus, so everything possible is in focus", or "focused on something infinitely far away". In photography, it is specifically the second of these — infinity focus means your lens is focused so an infinitely-distant object would be sharp.

When an object is infinitely far away, the rays of light coming from it are, in theory, parallel. A lens set at infinity focus is set so that theoretical object would be in focus.

In practice, all stars are close enough to infinitely far away to count, and in fact for most purposes for photography, everything beyond a hundred feet or so (depending on focal length) is as well.

Note that almost all modern autofocus lenses can focus past infinity. This allows for much wider tolerance in manufacturing, and makes it possible to allow for changes due to temperature. Older manual-focus lenses usually are made so turning the focus ring all the way puts it at infinity.

Some macro lenses (or macro configurations) do not allow infinity focus, and this is also the case with some situations where you're adapting one lens type to another mount.

For earth-bound photography, if you wanted that first thing — an infinite amount of focus — the hyperfocal distance is often more useful than infinity focus, as this maximizes practical depth of field. If you're looking for infinite depth of field, unfortunately, you can't really have it (not with traditional optics), but this and a small aperture will do in many cases.