This is a very broad question. You could get a better answer if you asked a more specific question.
The abolition movement in the United States was not very strong and not very popular. Most people, including Northerners, did not believe in abolishing slavery.
The movement really got started in the 1830s, due largely to a wave of reforms that started up in that era. Some major names in the movement were William Lloyd Garrison, Frederick Douglass, Lydia Maria Child and the Grimke sisters.
The abolition movement never really had much of an impact on government policy. They did, however, manage to help quite a few slaves escape from the South. Their main legacy, however, is that they spoke up against an evil system when hardly anyone was doing so.