What exactly are finger sprains?

Joint injuries. A finger sprain is an injury involving the supporting structures of a joint. The joint capsule and ligaments, provide stability to the joint. In a sprain, these structures are excessively stretched, buy not disrupted. Sprains resolve with time and can be helped by brief immobilization, protected range of motion, ice, nsaid's and steroids.
Ligament injury. A sprain is an injury to a ligament. These can be minor and require no treatment or can be severe requiring surgery.
Soft tissue. A sprain is a soft tissue injury to a specific joint, so a finger sprain is a soft tissue injury to a joint or joints in a finger. The soft tissue structures that are usually involved are ligaments, which connect one bone to another.