The Vulture is a mob added in the "Desert" update. It spawns in the Desert biome. Normally, it flies around like a hawk. However, when on the ground, if a player is nearby, it will continue attacking the player until the player or the vulture dies, if the player is under a roof, or if the player leaves the vulture's aggro range. The vulture and hawk are the only two mobs that can fly without a condition (the dragon and lava dragon will fly back to their lair when not in their own biome and no player is within their aggro range).
Vultures can be difficult to kill, due to their high health and speed. The easiest way is to hold a sword and attack slowly, even slower than you would kill a polar bear or arctic fox. If you take damage, temporarily let go of your weapon.
Alternatively, you can kill the vulture with a pickaxe. You are much less likely to get hit, but it takes longer.
Do not go under a roof when trying to kill a vulture, as it will fly away. If there is a sandstorm, and you have a turban, just keep fighting the vulture. You can survive by occasionally taking aloe vera. If you do not have a turban during a sandstorm, hide under a roof, and kill a vulture when the sandstorm ends. Do not go into the water, as the vulture will be significantly faster than you and make short work of you.