Basically, a natural arch is a rock formation with a hole in it.
If you're wanting a more formal definition, here's how the Natural Arch and Bridge Society defines a natural arch:
Outdoor enthusiasts of all ages and experience can visit many of the natural arches and bridges in Utah's national parks like Arches National Park, Bryce Canyon National Park, Canyonlands National Park, Capitol Reef National Park, and Zion National Park. Additional natural arches can be found in the Grand Staircase-Escalante National Monument, Natural Bridges National Monument, and Glen Canyon National Recreation Area.
If you are tired of doing the same old boring things, looking for something new and exciting, want an activity to do that is outdoors and is healthy for you, then try searching for natural arches and bridges.
Arch hunting is more than looking for holes in rocks; it is also the adventure along the way that makes this a worthwhile and popular hobby.