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When you say concierge, do you mean the type of cabin or the concierge service?
On the original ships (Magic and Wonder) the concierge cabins are the suites. On the new ships (Dream and Fantasy) the suites are concierge, but there are also some of the largest 'regular' verandah cabins (family verandahs) that are allocated to concierge service.
I've never sailed in concierge, but my understanding is that concierge services provide someone who can do things for you: so you never have to stand in line at guest services, or to make a dining reservation, shore excursion, etc. I think there is a complimentary lounge with snacks and drinks. On the Dream and Fantasy there are special concierge-only sun decks. If you are in a suite, you can order room service from the specialty restaurants. You are guaranteed your choice of dining (main or second seating, rotation, size of table, etc.).
I believe one of the big perks with concierge is the ability to book shore excursions, Castaway Cay cabanas, specialty restaurant reservations, etc. before anyone else, at a very early window. You also are the first to board the ship and have priority for disembarkation.