Renewing Subscriptions

Can someone please explain the difference of the mode details under the Membership Subscriptions please? I am a bit confused between: Finite, Infinite and Serial?

I had a member sign up but the billing cycle in my Paypal account shows the recurring cycle as zero. How can this be changed? On top of that after she set up the subscription then it sent her an email stating that her payment went through but the subscription was cancelled.

Julia