Collators create PoV blocks and provide them with candidate blocks from the parachian to the relay chain. So they don't do any of the block authoring since this is done by the validators.
But is there any incentive for someone to run a collator or is it only to support a specific parachain?
It depends on the chain.
It's totally up to you.
On parachain-template, the system will share some tx fee with collators. https://github.com/substrate-developer-hub/substrate-parachain-template/blob/664114fe707c089c112a2db770403cccf01c256d/runtime/src/lib.rs#L435
On Darwinia, we have darwinia-staking.
On Moonbeam, they have parachain-staking.
