I want to move and the landlord said I'm to give two month notice. I only heard of a one month notice.

Do you have a lease? If so then you need to read it. Whatever the lease requires is the amount of notice that you need to give. If you don't have a lease and you're a month-to-month tenant then customarily it's one month notice.

(If you had a lease that expired and never signed a new one, the notice requirements in that former lease would still be valid.)

It will all depend on your rental agreement, many written leases require a 60 day notice.

Also where you live may matter. I don't understand why someone would ask a question of law without giving the general area since laws vary.