The bee that leaves its barbed stinger in mammalian skin and dies afterward is the honey bee.
This description specifically refers to a worker honey bee. Its stinger has backward-facing barbs that anchor it in elastic skin such as human or other mammalian skin. When the bee pulls away, the stinger tears loose along with the venom sac, muscles, nerves, and abdominal tissue, fatally injuring the bee.
The detached apparatus is not immediately inactive. Its muscles can continue driving the stinger deeper and pumping venom, while alarm pheromones released near the sting site can attract other defensive workers. This is why removing a honey-bee stinger promptly is important.
The statement is not true of every bee in every circumstance. Bumblebees, carpenter bees, and sweat bees generally have smoother or less strongly barbed stingers and can usually sting repeatedly. Honey-bee workers can also sting insects without necessarily losing the stinger, because an insect’s hard exoskeleton does not hold it in the same way mammalian skin does. Male drones cannot sting at all.