Rick is certainly correct that the three-fifths compromise is typically misunderstood. I would like to add an additional clarification. Under Article I, Section 2, slaves–referred to as “all other persons” (that is, other than free persons, indentured servants, and “Indians not taxed”)–were to count as three-fifths persons for purposes of apportioning both political represenatation and taxation. Rick is correct that slave states wanted slaves to count as whole persons for purposes of political apportionment, but they also did not want slaves to count at all for purposes of apportioning direct taxes. Part of the compromise was connecting the two, so slave states would go along with the reduced representation.
While we’re on the subject, it is worth noting that, despite the text’s accomodation of slavery here and elsewhere, the document never makes explicit mention of the peculiar institution. Rather, slavery is always acknowledged indirectly, as with “all other persons.” This was no accident, for while a political accomodation was necessary to ensure ratification, many understood the existence of slavery was fundamentally at odds with the principles upon which the Constitution was based.