|The following is based on Fallout: Brotherhood of Steel and is not canon.|
He is the one who becomes a link between the townsfolk and the local raider gang, keeping both sides at least partially satisfied. Raiders get the slaves and supplies they need, while citizens mostly remain alive. When the Initiate arrives, he begins to despise the mayor's activities.
Later, the mayor decides to help the raiders kill the Initiate, who had recently defeated a group of them in the bar. In doing so, Richard hoped raiders would spare the citizens, and that things would generally remain the same. Richard knew the Initiate was looking for Brotherhood paladins, so he lied, telling the Initiate that they were going into Carbon crater.
When the Initiate made it to the bottom of the crater, Richard had already been waiting for him here, along with a bunch of raiders. Despite the raider's aid and great amount of explosives Richard used against the Initiate, he was ultimately shot. While dying, all the explosives he still had in his pockets detonated, blowing him to pieces and triggering the crater to collapse. 
Despite Richard's attempts to protect Carbon by killing the Initiate, the raiders ignored him and attacked the town.
Richard the mayor appears only in Fallout: Brotherhood of Steel.
- Fallout: Brotherhood of Steel Manual p.32: "Mayor Richard
As the town mayor, Richard isn’t entirely a bad man, just a regular guy beaten down by the times and desperate to maintain his small realm any way he can. Carbon may not be all that exciting to look at (or live in) but it is his town, and he’ll do anything to keep it that way."