Prime broker

A prime broker is the business division of an Investment Bank that looks after hedge funds.

It usually provides the following services:

Consulting: Helping a nascent hedge fund get off the ground: setting it up, finding offices, hiring people, engaging lawyers, recommending (cough) prime brokers, and capital introduction - the tricky business of putting investors and investment managers together without looking like it was anything to do with you.
Custody: looking after the hedge fund's "long" investment portfolio
Bank accounts: running a multicurrency cash account
Settlement facility: lending the hedge fund the assets it needs to settle short sales, and lending it the money it needs to buy assets.
Synthetic prime brokerage:providing the hedge fund with exposure to assets through derivatives (as an alternative to the hedge fund buying them outright). Often in the trade called "CFD"s. This may involve accepting give-ups from other executing brokers.
Swaps and ETD': Providing general exposure to swaps, futures, options and that sort of thing.

Act as PB administrator: While they look after assets, prime brokers don't calculate NAV (that's the PB administrator's job)
Act as depositary: Nor do they act as an official depositary for AIFMD purposes (though they may get delegated the safekeeping rols and may act as a depo-lite)
Act as an executing broker: They don't themselves work equity orders for their clients (though their compadres across the Chinese wall in the equities trading division at the same investment bank almost certainly will)

See especially in the context of AIFMD: Prime broker - AIFMD Provision.