You've got to obtain an order for possession from the Court. This is an Order from a Judge which requires the tenant to vacate. If they do not then you can ask for a warrant for exection from the Court so that their Bailiffs remove the tenant.
If you have a written tenancy agreement for an assured shorthold tenancy agreement and a copy of the notice then you can use the acceleratede possession procedure so that the order is made without the need for a formal Court hearing. You will have to complete a form and submit it to Court and I attach the relevant link to Her Majesty's Courts Service page for the form:-
You will also see that there are some guidance notes on the form. If you run in to difficulty you can just call your local Court about completing the form - they are usually pretty helpful, particulaly with lay persons.
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