I have registered a domain name and created an email account for that domain using Google Apps. I want to know what will happen to my Google Apps account associated with that domain after its registration expires or if I’m not going to use it (the domain I mean).
When the domain name expires so will the MX records for that domain. As you have already verified the domain with Google theoritically you will still be able to send email as that domain (since you are using Google servers to send) however you will no longer recieve incoming mail.
Its possible that Google might detect the domain is no longer registered and prevent you from sending mail.
That being said, Google wont disable or delete your apps account when your domain expires (provided you continue to pay the subscription) - all your data will remain and the account will still function, just without email (eg. you will still be able to use your Google docs)
Under your billing information you will see exactly what you paid and for how long. If you're going to renew it after a year then you don't have to do much, but if you don't then it expires.