In very general terms, RAIM (Receiver Autonomous Integrity Monitoring) and FDE (Fault Detection and Exclusion) are means to provide a GPS receiver with a cross check function and exclusion of an unreliable satellite for navigation purposes.
In VOR navigation, there is a flag on most CDIs that will warn pilots when a VOR signal is not valid or not available. RAIM provides the means for the GPS receiver to indicate a problem with a satellite. FDE then checks each satellite against each other and exclude the failed satellite when a problem is detected.
RAIM is always required in GPS receivers that are certified to TSO C129A or better. This certification, in most cases, allows the GPS receiver to be operated and utilized for IFR navigation.
If a GPS receiver has the FDE feature, then, in most cases, it may be used as the sole navigation source in oceanic and remote flying operations. Information such as dates, times, route, waypoint, speed, etc is entered by the pilot into the GPS or FMS to make the necessary FDE prediction to determine sufficient satellite availability, integrity, and accuracy for the FDE flight plan. These predictions are made possible due to the known, published orbits of the satellites around the Earth. That is, if the GPS knows where each satellite is going to be located in the sky on each part of a flight plan, then it can make the necessary adjustments and exclusions on which satellites to use or not use.