CCD-IK (Cyclic Coordinate Descent)
A classic iterative IK solver that walks joint by joint up the chain, rotating each one so the end lands as close to the target as possible.
Cyclic Coordinate Descent works one degree of freedom at a time. It starts at the joint nearest the end, rotates it so the vector from the joint to the end points toward the vector from the joint to the target, moves on to the next joint up and repeats the whole pass until the end is close enough. Each individual turn is a simple angle problem, so the method is cheap and easy to write.
The idea comes from robotics — Wang and Chen described it for manipulators in 1991 — and was quickly taken into skeletal animation. It converges fast but tends to over-rotate the outer joints, giving slightly unnatural poses unless per-joint angle limits are added. In games it has largely been joined by FABRIK, which moves points instead of turning angles and gives smoother results.