The simple definition of an arc shot is a shot where the camera circles its subject. In an arc shot, the subject is typically still and the camera provides the motion, tracking around the subject in at least a semi-circle of movement. (The term comes from mathematics, in which "arc" has the same definition as segment of a complete circumference.)
Film analysts tend to cite the prom dance scene from Carrie (1976) as a chief example of this shot in use. The dancing is relatively stationary as the camera circles the actors from a low angle, capturing the monument of the occasion for Carrie (Sissy Spacek) and reflecting her inner joy.
By classification, an arc shot is typically also a tracking and/or dolly shot, as the camera is moving on its subject. This simple video depicts a basic arc shot on a wooden figure:
An arc shot is often used to reveal different components of the area in which the subject is standing. At the start of an arc shot, the area behind the camera operator is the visual background by the time the shot finishes, allowing for greater depth and detail about the subject's surroundings.
Shooting a slightly moving subject, like the dancing couple in Carrie or the soldier in Black Hawk Down (2001), makes for a visually complicated and engaging shot -- moreso than utilizing an arc shot on a fully stationary object. It adds a bit of drama.
The challenge of an arc shot lives in the need for a constant rate of movement while circling the subject at a fixed distance. It is challenging to do free-hand, and a dolly is often utilized to maintain consistency. However, that rule of consistency is not concrete. Arc shots occasionally move in on or away from their subject, and can maneuver height, while circling if the intention of the arc shot is in revealing different places of the subject, such as this bullet-dodging arc from The Matrix (1999), which begins about 38 seconds into the following clip.
It is worth noting, however, that no arc shots in The Matrix films were done with a single dollying camera, but with multiple cameras filming from many angles that were edited together. This creates an arc effect through different practice.
Finally, this interpretation from Lost In Space (1998) has two exaggerated arcs happening at once, cut back and forth between one another (beginning around 1:22 in the below video).