A tenant can move out during a fixed term tenancy if their tenancy agreement contains a break clause, stating that either party can end the tenancy early (ie before the term has expired). If there is no break clause in the agreement, then you can only end the tenancy if both parties agree to it. This is called surrendering the tenancy.
Where there are joint landlords (ie more than one landlord is named on the tenancy agreement), you should generally agree with all landlords to end the tenancy early. However, where one joint landlord acts as an agent for the other(s), they can accept a surrender on behalf of all the landlords. If joint landlords have appointed an agent to act for them, and you have been instructed to deal with this agent, you can deal with the agent to end your tenancy early.
There are two types of tenancy surrender - express and implied surrender.