Kalman Filters
A Kalman Filter in the context of a flight management system is a piece of software that allows the FMS to combine multiple different sensors to determine a very accurate position and velocity. It allows the system designers to combine inputs that have quite different error characteristics in a sensible manner. For example, modern FM systems will often have air data inputs (airspeed, altitude, rate of climb), heading inputs, the ability to tune and sense a VOR, the ability to tune and sense a DME, an input from whatever inertial platforms are on board (AHRS or whatever), and an input from one or more GPS sensors. Each input will be processed according to its' inherent error characteristics, and will generally be allocated a weighting - how strongly it influences the position and velocity solution. The cool thing about a Kalman filter is that it automatically and continuously updates these weightings based on current performance of each sensor. That's why it is a filter.
The previous post mentioned Kalman Filters for correcting IRS errors. That is one use, but they are more widely used in modern FM systems for multi-sensor blending.
Some FM systems have the ability to display what weightings are in force at any particular time. In our FMS, if all is well with the world, you can see that it is heavily oriented to the GPS input. If the GPS input is disrupted, it seamlessly drops the priority of the GPS and uses DME/DME scanning. On a really bad day it will eventually end up in dead reckoning mode, in which it is using airspeed and heading to estimate where you are.
But, even when the GPS is working perfectly our FMS never weights 100% to the GPS position source. It is always running all the other inputs, because that is how it can spot a GPS outage very quickly, plus the AHRS updates are quicker, so that in manoeuvring flight the FMS position is often a few metres different to the GPS position - and correctly so.