Emancipation today is most frequently used to refer to the emancipation of a child (not necessarily a minor) from their parents. The key legal effect of emancipation is that the minor's parents are no longer required to financially support their child.
Generally, emancipation occurs automatically at a defined age (usually 21 in the US), but it may occur earlier if the child voluntarily leaves home and withdraws from the guidance and control of their parents.
Going to college, for instance, would not generally constitute emancipation. Usually, the child gets money from their parents to attend college, and is forced to put up with their parents' guidance as to study, diet and sexual relations. However, if the child gets a job to pay their own way through college, and generally stops listening to their parents' advice, they may be considered to have emancipated themselves.
One common ground for early emancipation is joining the military, since a person in the military is under the total control of their commanders and completely outside the control of their parents.
Besides cutting off the obligations of custodial parents, emancipation also cuts off a non-custodial parent's obligation to pay child support. (Caveat: make DAMN sure your child is legally emancipated before stopping child support payments, because jail sucks.)