Short answers, pulled from the story.
Hungarian émigré Rudolf E. Kálmán published a technical paper introducing the core algorithm in 1960. Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory during the early 1960s, leading to the joint designation Kalman, Bucy filtering.
The Kalman filter combines physical laws of motion with noisy GPS measurements to produce a better estimate than either source alone. It assigns greater weight to estimates with higher certainty while giving less weight to those with higher uncertainty through recursive calculations.
The extended Kalman filter handles nonlinear functions by computing a matrix of partial derivatives called the Jacobian at each timestep. The unscented Kalman filter uses deterministic sampling techniques known as the unscented transformation to pick sigma points without requiring explicit Jacobian calculations.
Kalman filters have been vital in implementing navigation systems for U.S. Navy nuclear ballistic missile submarines since their inception. They guide cruise missiles such as the U.S. Navy's Tomahawk missile and the Air Force's Air Launched Cruise Missile.
Some equations appeared in papers by Soviet mathematician Ruslan Stratonovich before summer 1961, when Kalman met him during a conference in Moscow. This meeting occurred after Kalman published his initial technical paper in 1960.