During the sign up process you will be sent a verification email by Google. You will need to follow the instructions in the email to complete the sign up process. Once you have completed the sign up process it can take up to 72 hours for your Google AdSense account to be verified.
If you already have a Google AdSense account and have just added a new domain name to it, after you have configured the ads for your domain name, you will be sent an email to verify the association of the domain name to your Google AdSense account. Follow the instructions on the email and allow for up to 72 hours for the ads to be displayed on your website.