Backpacking (also tramping or trekking or bushwalking in some countries) combines hiking and camping in a single trip. A backpacker hikes into the backcountry to spend one or more nights there, and carries supplies and equipment to satisfy sleeping and eating needs.
A backpacker packs all of his or her gear into a backpack. This gear must include food, water, and shelter, or the means to obtain them, but very little else, and often in a more compact and simpler form than one would use for stationary camping. A backpacking trip must include at least one overnight stay in the wilderness (otherwise it is a day hike). Many backpacking trips last just a weekend (one or two nights), but long-distance expeditions may last weeks or months, sometimes aided by planned food and supply drops.
Backpacking camps are more spartan than ordinary camps. In areas that experience a regular traffic of backpackers, a hike-in camp might have a fire ring and a small wooden bulletin board with a map and some warning or information signs. Many hike-in camps are no more than level patches of ground without scrub or underbrush. In very remote areas, established camps do not exist at all, and travelers must choose appropriate camps themselves.
In some places, backpackers have access to lodging that are more substantial than a tent. In the more remote parts of Great Britain, bothies exist to provide simple (free) accommodation for backpackers. Another example is the High Sierra Camps in Yosemite National Park. Mountain huts provide similar accommodation in other countries, so being a member of a mountain hut organization is advantageous (perhaps required) to make use of their facilities. On other trails (e.g. the Appalachian Trail) there are somewhat more established shelters of a sort that offer a place for weary hikers to spend the night without needing to set up a tent.
Most backpackers purposely try to avoid impacting on the land through which they travel. This includes following established trails as much as possible, not removing anything, and not leaving residue in the backcountry. The Leave No Trace movement offers a set of guidelines for low-impact backpacking (Leave nothing but footprints. Take nothing but photos. Kill nothing but time.).