Generally, when a landlord give notice to quit/cure, you have within that time (in your case 15 days) to make good. If you have the rent before then, then you are fine because the landlord must accept it. However, after the 15th day, the landlord does not have to accept your rent money. However, you don't have to move out right away. The landlord must file a eviction lawsuit and get judgment before they can lawfully evict you. Depending on where you live, you may have anywhere between a week or so to one or two months additional time. In any event, it is not a criminal case, so they cannot go to the police to kick you out until a judgment is received.
As for the notice, if the lease is already finished and you are on a month to month basis, then the landlord can give you notice to move out regardless of whether you are paying rent or not. The amount of notice defers depending on which State you live in. Generally, in California, its about 30 days.