Absent an agreement the default is a month to month tenancy under the Residential Landlord Tenant Act. You need to provide a 20 day notice to terminate at the end of a month. As an example - you could give a 20 day notice to vacate by July 31 in the next week or so and if the tenant failed to move out on July 31 you could start an eviction action on August 1. The 20 day notice needs to be handed to the tenant or posted and mailed with proof of service.
You may find this website helpful.
If the tenant has been paying rent, I agree with the previous poster. If the tenant has not been paying rent or never has paid rent, then they would be considered a tenant at sufferance or tenant at will. In order to get rid of a tenant at sufferance or will you must provide them with a 3 day quit notice. If they do not move within 3 days you may then start the eviction proceedings.
Again this all depends upon whether or not there has ever been rent paid by the tenant.
Tenancies at will or sufferance don't fall under the RLTA and don't require any particular notice. I respectfully disagree with the previous posters. Just tell this person the free ride is over and you need them to go.