How long does someone have to live in your apartment before you can legally tell them to leave?

If you paid rent for food for the two of you will probably be considered to be a "tenant"; here since there was no lease it would be a month-to-month tenancy.  I you didn't pay rent, etc. you will be considered to be a "licensee"; that is someone who entered and remained on the premises with permission. 

In either event, now that such permission has been revoked the lawful way to remove you from the premises is to serve him with a notice to quit (typically30 days).  If you fail to leave at the end of that time she will have to go through the requisite procedures for eviction in your state.  Once the court enters an order for you to vacate the property it will be enforced by the sheriff, using physical force if necessary.