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McPherson, Battle Cry of Freedom, covers United States conscription from near the start of chapter 16.
Recruiting of volunteers and calling up of militias had served their needs for 1861. But after US failure in the Seven Days', morale plunged for a time. Recruiting went less well in the new call of July 1862. "The parades and rallies of 1862 were a pale imitation of the stirring demonstrations of 1861. The lengthening casualty lists had taught people that war was not a glorious game. Although the North had mobilized only a third of its potential military manpower, a booming war economy and the busy summer season on farms had left few young men at loose ends and ready to volunteer. Moreover, the new recruits, like those already in the army, had to enlist for three years." I presume that recruiting went badly in Democratic areas like New York City and the butternut areas of southern Indiana and southern Illinois. The later draft certainly did (the US sent troops into four states to keep order and carry out the draft).
His discussion of the militia act of 17 July 1862 is confusing, not least because the act itself is confusing. State militia men were called up for up to nine months and the War Department was enabled to start levying. There were formulae where missing volunteers under an earlier call caused levys of militia men (I gather there was districting and quotas per district), but "every three-year volunteer enlisted above quota would be counted as four men against the nine-month militia quota." The purpose, expanded on later, was more to stimulate volunteering rather than to cause a draft.
Ch. 20: a further draft act became necessary in 1863. "The men likely to enlist for patriotic reasons or adventure or peer-group pressure were already in the army." More war weariness and more of a booming economy. "The still tentative enlistment of black soldier could scarcely begin to replace losses from disease and combat and desertion during the previous six months. Like the Confederacy in early 1862, the Union army in 1863 faced a serious manpower loss through expiration of enlistments ..."
Quotas were assigned to districts (congressional districts). "Each district had fifty days to fill its quota with volunteers [those who had previously volunteered counted too]. Those that failed to do so then held a lottery draft to obtain a sufficient number of men to meet the quota."
But he adds, "If a man's name was drawn in this lottery, one of several things would happen to him next---the least likely of which was induction into the army."
Result: of the 776K called up in the first four drafts, only 6% (46K) were drafted into the army.
Lots of cities, counties, and political machines paid the commutation fees for men. Some businesses paid for commutation based on wage levies. You could buy actual draft insurance policies.
"... it was not conscription at all, but a clumsy carrot and stick device to stimulate volunteering. The stick was the threat of being drafted and the carrot was a bounty for volunteering. ... [reciting the draft and substitute figures], some 800K men enlisted or re-enlisted voluntarily during the two years after passage of the conscription act."
Some bounties were given by the War Department. There could also be benefits in choosing your own regiment or electing its officers. States, counties, and cities provided bounties, to the soldiers or to help support their families. "... the shock of the first draft, which provoked bitter resistance in many areas, caused communities to resolve to fill future quotas by any means possible to avoid a draft." There could be massive piles of bounties late in the war -- $1000 or more.
But he concludes the bounty men and sustitutes, though technically volunteers, didn't really contribute a lot to the victory, due to desertion (with optional repeated bounty hunting). He writes that the original volunteers of 1861 and 1862 mostly led the US victory.