Regarding a venture capital (or private equity) fund organized as a limited partnership, the carried interest would be the general partner's share of the capital gains generated through the fund's performance (via liquidity events with respect to portfolio investments). Carried interest does not refer to the management fee charged by the venture capital or private equity general partner.
Typically, a fund must return the capital invested in it by limited partners (plus any agreed hurdle rate or preferred rate of return) before the general partner can share in the gains of the fund. The general partner will then receive its agreed carried interest (also known as its "carry"), which is the agreed share or split (with the limited partners) of the remaining gains. The carried interest is the general partner's principal financial incentive to perform well and generate strong fund returns.