What rights does a landlord have with regards to eviction when he/she has allowed a relative to live rent free in a home owned by landlord for years without any form of lease agreement.

You have to evict according to the laws of your state. If you want to tell us the state, we will tell you, or you can ask at your town hall or county court. You will start by giving the person a notice to vacate the premises, in writing. Generally this is 30 days minimum, but in some locales it might be 60 days. If you can tolerate 60 days, do that now. Do it in person with a witness. If you are far away, send it certified mail.

The same rights as anyone who allows someone to live in their property. The only difference is that, absent an agreement to pay rent, you cannot use non payment of rent as grounds for terminating the tenancy.

They will be considered a periodic (month to month) tenant in the eyes of the law. The general process is a written notice to vacate with a deadline (30-60 days). If they do not leave by the deadline, then you need to go to the appropriate local court and apply for an eviction order. This means a hearing and then they are given a short amount of time to vacate. If they still refuse, you hire a sheriff to physically remove them.