First you choose which permanent you'll sacrifice (if you control any of the chosen type), then each other player in turn order does the same, then all chosen permanents are sacrificed at the same time.

You don't choose a card type until the ability resolves. Once you choose a type, it's too late for players to respond.

You may choose any card type. Choosing anything other than artifact, battle, creature, enchantment, kindred, land, or planeswalker will result in no permanents being sacrificed, though, since these are the only types permanents can have.