Can someone help me understand this sentence in my lease agreement?
Basically it means that you must give no less than 30 days' notice unless the landlord-tenant act applicable to your lease requires more notice, in which case thee longer time applies to the notice requirement. You'd have to refer to the applicable landlord-tenant law in your jurisdiction to determine whether this is the case. As for your question about the remainder of the lease, without seeing the full lease it's impossible to say; you should seek the advice of a licensed attorney in your jurisdiction familiar with residential leases, or advice from a tenants' rights organization in your area.