A green address is a special trusted Bitcoin address that is used to indicate the origin of funds to a recipient. Assuming the recipient trusts the owner of the address to not attempt a double spend, the recipient may treat the funds as confirmed the moment they arrive.
As an example, assume website Z accepts incoming Bitcoin payments, but also trusts the green address published by Mt. Gox. Customer C wants to withdraw funds from Mt. Gox and send them to a payment address of website Z. A customer who does a withdrawal from Mt. Gox can click the use green address checkbox, which will result in the payment being sent to the green address as an intermediate step before forwarding the payment to site Z. Site Z can confirm that the payment passed through Mt. Gox's green address and trust the payment as confirmed immediately, since it knows that the only party who could potentially perform an attack to reverse the payment is Mt. Gox, which is presumed unlikely.
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Green addresses are generally considered to be a bad idea because they encourage transaction spam. This position is held by most, if not all, of Bitcoin developers.