It is relatively easy to pinpoint the date of the ending of the abolitionist movement, but it is very difficult to identify any date as the beginning of that movement.
It is fairly easy to say when the abolitionist movement ended even though there was no official ending to it. The movement was clearly completely finished when the 13th Amendment was ratified and slavery was officially made unconstitutional in the United States. This happened in December of 1865. The abolitionist movement was no longer necessary as such in the United States because slavery no longer existed. Some abolitionists moved on to advocating for other issues such as black rights, but this was no longer an abolitionist movement.
As to when the movement began, it is much harder to say. There was a meeting of Quakers who met to protest slavery as early as 1688. Thus, you could say that the abolitionist movement existed from that date. There were some abolitionists working against slavery in the early 1700s as well.
However, the main American abolitionist movement probably can be said to have begun with the Second Great Awakening. This was a religious revival movement that peaked in the 1820s and 1830s. It promoted the idea that people should work to perfect their society. Some Americans took that to mean that they should work against slavery. It was during this time that famous abolitionists such as William Lloyd Garrison became prominent.
Thus, it is probably best to say that the abolitionist movement lasted from some time in the 1820s to about 1865.