In the world of finance, carried interest is the share of the profits in an investment that a manager (of said investment) earns in excess of what they may have contributed to the partnership. For example, let's say that a manager is putting in 10% of the cash that is required for a particular project. If the project goes really well, the manager, through carried interest, could earn more than their 10% share of the profits. Put another way, it is a performance fee that is intended to incentivize and reward the manager.
Today I learned (credit to Lucas Manuel) that the origins of carried interest go all the way back to the Middle Ages. The concept and term supposedly came about because the captains of European ships would take a share of the profit from the "carried goods" that they were transporting. This was to compensate them for the work and for the risk of sailing all over the place. Keep in mind that, just like today, any number of things could have gone wrong. Maybe you don't make it or maybe pirates steal all of your goodies.