The non persistent cookie will live in user browser. the moment user will close the browser the cookie will be deleted. This is the normal way we use with authentication cookies.
persistent cookie on the other hand get saved on the user hard disk. this is typically done when we want to implement "remember me" type of lo-gin functionality. since the cookie will be saved on user computer, next time when the user access the page that cookie will serve as authentication ticket and the user will be logged in.
As for why is it encrypted, if we don't encrypt it then we are potentially sending user credential related sensitive information over the internet. It can be eavesdropped and then user maliciously by someone else (perhaps a hacker or hacker wannabe)
You can refer to following article for details on custom forms authentication: Understanding and Implementing ASP.NET Custom Forms Authentication