Annihilator is a keyword ability that appears on 10 Elrdrazi cards from Rise of the Eldrazi. Annihilator is a triggered ability that triggers when the creature with Annihilator attacks. When this triggered ability resolves the defending player chooses and sacrifices permanents equal to the number after the Annihilator keyword ability.
As this ability triggers when the creature with Annihilator attacks, the ability will resolve in the declaration of attackers step before blocking creatures are declared. This means that you cannot block with a creature and then sacrifice it to the Annihilator ability.
If the attacking creature with Annihilator is attacking a planeswalker and the planeswalker itself is sacrificed to the Annihilator ability, the attacking creature can still be blocked, but if it is not the creatures attacking this planeswalker will not deal any combat damage.
In a two-headed giant game, the controller of the creature with Annihilator will choose which player will sacrifice permanents when the Annihilator ability resolves.