It worked like this in the Union:
1. President calls for troops, and sends quotas to each state's governor.
2. Governor selects leading men, military or otherwise, and gives each man a temporary colonelcy and the authority to raise a regiment in a certain area (usually representative districts) within a certain amount of time (usually 2-3 months).
3. Temporary colonel selects a recruiting station in a town in the area (or a building if in a big city) and puts out posters. If he's a militia colonel, he might call on members of his militia unit to enlist. If he's a prominent politician, he might call a Town Hall or other big meeting to encourage enlistments.
4. When Temporary Colonel gets enough volunteers, the state recruiting director (usually a Regular Army officer on detached duty) shows up and musters the men into federal service.
5. Governor appoints officers (usually line officers in already-serving units, sometimes Regular Army officers with impressive records, and sometimes the Temporary Colonel if he's done a good job), and regiment goes off to war.
(Alternate 4: If Temporary Colonel reaches the time limit and doesn't have enough volunteers, the number of men he has is folded into a different Temporary Colonel's recruiting effort. The best example of this is the 178th New York Infantry, which took about a year to form and was the product of combining twelve unsuccessful recruiting efforts. Only one of the twelve Temporary Colonels got a commission.)