Unless the lease or rental agreement specifies otherwise, in most states there is no legally recognized grace period -- in other words, if a tenant hasn't paid the rent on time, the landlord can usually deliver a "pay or quit" notice the day after it is due. However, some leases and rental agreements do provide a five-day grace period for late rent.
Similarly, there is no legally recognized day that rent is due. Often rent is due on the first day of the month, but it is legal for a landlord to require rent to be paid on a different day of the month or at intervals other than monthly.
Some landlords charge fees for late payment of rent or for bounced checks; these fees are usually legal if they are reasonable. The laws on late fees can be found in your state's landlord-tenant statutes.
For everything you need to know about rental agreements and leases, get Every Tenant's Legal Guide, by Marcia Stewart and Janet Portmant (Nolo).