If my mother evicts me from her apartment, do I have a certain amount of time to leave or must I leave immediately?

There is no simple answer. You say you do not pay rent, but you do pay the internet bill. It is possible that a court could conclude that IS rent. If so, you would be a month-to-month tenant on an oral lease, and she could give you 30 days notice terminating your tenancy; i.e. ask you to leave on 30 days notice. But if a court did not consider that rent, then you are a guest, and she could ask you to live immediately, without notice. Certainly, you'd want to argue that the payment of internet is your rent, and it's a reasonable position to take, but it is far from guaranteed that you would prevail on it; that's why it's impossible to give a firm answer as to how much notice you are entitled to.

In either even, if you do not move out when you are told (to either immediately, if a guest; or on 30 days notice if a month-to-month tenant), then your mother could file a court action to remove you (the actions differ slightly based on whether you are a guest or tenant, but for simplicity's sake, we'll call both an "eviction" action). Your mother may NOT simply change the locks, throw your belongings on the street, etc.--the only way to evict a person who had been living there with permission or as a tenant is through the courts. So presumably, you'd in effect have a month plus of time even after you were supposed to go--the time between her filing the legal action and it going to court. However, you'd want to leave before the trial date and get the case dismissed, since you would not want an eviction on your record when looking to rent elsewhere.

Also, your mother would have to give you an opportunity to recover/retrieve your belongings or else ship them to someplace you designate (though she could seek any storage or shipping costs from you); she may not keep them or throw them out.