The best-known compromises that were made relatively soon after the United States became independent were the compromises that were made as part of the writing of the Constitution. The two main compromises in the Constitution were the Great Compromise and the Three-fifths Compromise.
The Great Compromise created a bicameral system with different ways of apportioning seats in each house. It gave the small states some of what they wanted because it created a Senate where all states had equal representation. At the same time, it gave the big states some of what they wanted because it created a House of Representatives where the big states had more seats.
Since the House was apportioned on the basis of population, the Three-fifths Compromise became necessary. This compromise was between slave states and free states. The slave states felt their slaves should be counted as population for the purpose of apportioning seats in the House. The free states did not think this made any sense. The compromise was to count 3/5 of the slave population for the purposes of apportioning seats in the House.