Raleigh was the capital of North Carolina. It was the home of the Governor of North Carolina, who rarely visited Mayberry, and a place to which Barney Fife both vacationed and eventually moved.
Whenever Barney went to Raleigh, he always wrote letters back to Andy, explaining how much he was enjoying once again staying at the YMCA, what "risqué" places he found there, and how good the breakfasts were.
As the home of the state police, "Raleigh" was somewhat personified by them, or others in the state bureaucracy. It was clearly depicted, in episodes like season 1's "The Manhunt", that Raleigh had the power to override Andy Taylor's authority as sheriff. However, this and other episodes consistently showed that Andy was able to show his worth to Raliegh-based officials, while remaining subordinate to them.
In real life[]
At the 1960 census, the population of Raleigh was 93,931. It is the capital city of the State of North Carolina.