Carpet beetles are very tiny little beetles, of the order Dermestidae. The most commonly found form of this insect is its larvae, which during their last larval molt, tend to climb up flat surfaces. In homes, this usually means the walls, and they're usually fairly conspicuous. They appear as a small grubs with brown and white rings, lined with hairs. These hairs, also known as urticating hairs, can break off if you touch them with bare skin, and can cause some itching and irritation. The adults, on the other hand, are very small beetles, with an attractive brown, white and orange mottled pattern, created by a series of colored scales on their backs. The adults do not cause any damage directly, as they tend to feed on primarily pollen.
The larvae however, can be a problem. Dermestid larvae have diverse tastes in food. Although, true to their namesake, they eat carpet fibers, they also have a taste for wool, fur and other animal based natural fibers, as well as some plant based fibers. They also have a tendency to eat away at cartilage, and dried flesh, making them useful to people making skeletal models of animals, as they can quickly clean a carcass down to a skeleton. This however makes them a hated enemy of people keeping taxidermied animals, and especially people who maintain insect collections. (An unfumigated collection of insects rapidly turns to dust and frass in a very short time.) In addition to animal products, the larvae also have a taste for flour, baking mixes and other bulk stored food, joining a host of other larder invading beetles, like grain weevils, and grain beetles.
They are also apparently extremely difficult to get rid off. For example, if they enter your mattress, you can't kill them from the outside. Or they go under your carpet, or behind a baseboard. Typical pest control doesn't help, meticulous housekeeping is the answer. Some people are unfortunate enough to move into an apartment with these guys, and there isn't much that can really be done, especially as they're so small and unnoticeable, unless population density becomes high enough to force larva to expand their diet. They can be quite common in areas where people frequently move, and a few could probably hitch a hide and start to live in your new home with you. Having one giant Moving Day probably doesn't help. In theory, if it was a piece of specific furniture, like a couch that was infected in particular, you could fumigate it, and it would in theory, kill the larva. However fumigation is both expensive, and dangerous. In a sense, this pest invasion is, besides fleas, one of the malignant side effects of the trend of wall-to-wall carpeting.
On an interesting note, relatives of the carpet beetles seen here are often found on dried corpses of animals, as well as humans, eating the hair, dried tissue and cartilage. As a result, they may be forensically significant in determining time of death on bodies. In particular, they are some of the last visitors to a corpse, as they prefer to hang around something dried out, as opposed to a messy, smelly and wet corpse. They help finish the jobs that flies started, in the last removal of dried tissue, cartilage and hair, and transforming the corpse finally into a relatively clean skeleton.