Theoretically, the Kalman Filter is an estimator for what is called the linear-quadratic problem, which is the problem of estimating the instantaneous “state” […] of a linear system perturbed by white noise by using measurements linearly related to the state but corrupted by white noise. The resulting estimator is statistically optimal with respect to any quadratic function of estimation error.
Practically, it is certainly one of the greater discoveries in the history of statistical estimation theory and possibly the greatest discovery in the twentieth century. It has enabled humankind to do many things that could not have been done without it, and it has become as indispensable as silicon in the makeup of many electronic systems. Its most immediate applications have been for the control of complex dynamic systems such as continuous manufacturing processes, aircraft, ships, or spacecraft. To control a dynamic system, you must first know what it is doing. For these applications, it is not always possible or desirable to measure every variable that you want to control, and the Kalman filter provides a means for inferring the missing information from indirect (and noisy) measurements. The Kalman filter is also used for prediction the likely future courses of dynamic systems that people are not likely to control, such as the flow of rivers during flood, the trajectories of celestial bodies, or the prices of traded commodities.
Yay! The above is the opening paragraphs of Mohinder S. Grewal and Angus P. Andrews Kalman Filtering: Theory and Practice Using MATLAB, Second Edition (2001, John Wiley & Sons, Inc.), which currently is under my scrutiny.