Normally, a five-day notice is served on you to give you five days to cure any breach, such as to pay all past due rent.
If you do not cure the breach within those five days, the landlord must bring you to eviction court to obtain an order of possession and monetary judgment, if any. You can stay in the unit until the eviction process is complete, which could take 30-60 days, depending on many variables.
If possible, you can also try to negotiate with your landlord to come to mutually agreeable terms to stay. Your landlord does not have to negotiate with you, but may, depending on the circumstances.
Disclaimer: This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.