A Notice of Forfeiture is used when the landlord wishes to evict the tenant due to the non-payment of rent or the breach of a term of the lease.
A Warning or Notice to Remedy Breach is used when the tenant has not paid their rent on time or they are in breach of a term of their lease and the landlord wishes to give them a period of time in which to remedy the breach before terminating the lease.
A Notice to Quit is used when ending a periodic lease, or when you have an overholding tenant. An overholding tenant is one that has stayed beyond the date specified to leave in a demand for possession.