The insect whose barbed stinger can remain in a victim's skin after a sting is the honeybee, specifically a worker honeybee. Its sting has backward-pointing barbs that anchor it in the thick skin of mammals. When the bee pulls away, the stinger and part of the abdominal tissue are torn from its body, and the worker usually dies soon afterward.
The detached stinging apparatus can continue pumping venom for a short time because muscles and nerves remain attached to it. This is why quickly removing a honeybee stinger is recommended. Honeybees generally sting defensively, especially when their colony is threatened; a foraging bee away from the hive is less likely to attack unless handled or crushed.
The rule has exceptions. A queen honeybee has a smoother, less strongly barbed stinger and can sting more than once, while a worker may sometimes pull free from thin or flexible skin. Hornets, wasps, and bumblebees usually have smoother stingers and can sting repeatedly without leaving them behind.