The precise manner depends on State Landlord-Tenant Law and the County Court, the latter of which conducts eviction hearings.
Since you have not specified your state, I can only provide a general answer.
Evictions can only be done by a court order, and they are executed by court-appointed officials- usually a county sheriff's officer. All tenants have a legal right to a court hearing, and the landlord must prove "just cause" for the eviction. The most common reason for a eviction is non-payment of rent.
The property owner/landlord must first issue the tenant a notice. This is called either a Notice to Vacate, a Notice to Quit, a Notice to Cure or Quit (this notice specifies what action the tenant must take in order to avoid being taken to court, or they must leave of their own volition.) The term of the notice must be equal to the that of the rental (7 days for a weekly rental, 30 days for a monthly rental). It must be issued before or on the 1st of the month/week. If you issue it on the 2nd, it only becomes effective on the following 1st.
Once the period of the notice has passed and the tenant has not left, then you can file for an eviction in county court. In cases of non-payment of rent, you may be able to file immediately.
In court, explain your situation. If you are granted the eviction, it can be anywhere from a couple of days to weeks before the eviction is executed. The tenant(s) and their property are removed and they are issued a no-trespassing order. If they return, they are arrested. In the case of a rental unit, as opposed to an owner-occupied rental, the door will be padlocked and a no-trespassing order posted on the door.
The precise procedure will possibly vary from the above- check State Landlord-Tenant Laws as well as the procedure in county court; a telephone call to a court clerk should suffice, but some courts post the steps online and even the court papers to file.