(Mining) The process that verifies and adds new transactions to the blockchain for a cryptocurrency. The miner that wins the competition is rewarded with the transaction fees.
Hosted mining occurs when a user has their cryptocurrency mining equipment hosted by a third-party provider. The provider takes care of all setup and maintenance required with mining (all for one set monthly fee) while the user receives full control over their equipment remotely. And because of that, cryptocurrency is never touched by the third-party provider, greatly reducing any risk of fraud or theft.