The three-fifths rule was a rule that was created to solve the problem of representation in the US Congress. With this rule, slaves were considered 3/5 of a person for purposes of representation and direct taxation.
The main arguments that led to this rule was that Southern slaveholders wanted slaves to count for the purposes of representation, because then the Southern states would be considered more populated and it would cause a bigger representation of their interests in Congress. But Northern states feared that this would give southerners too much power.
The main consequence of that rule was that with it Southern States held the balance of political power.