Before you sign up for a Google Adsense Account, please make sure that you have a suitable website with enough content that has never been applied for Adsense before. In term of suitability, your site should be free from porn information and adult content. Apart from that, you site should not be made solely for displaying adsense ads. Anyway, just read the TOS and see whether your site is eligible. In other words, your site should not be content-free.
Then go to Google Adsense Signup page: https://google.com/adsense/g-app-single-1, and start applying in AdSense. Provide your information including mailing address, your site URL etc. For those who wanna receive check, you can change your mailing address anytime, no worry.Afterwards, Google need to review your site and decide your site whether is suitable for putting adsense or not.
Normally, it only takes you 10 business days. However, for certain occasions, it will take as long as a month or so for Google to review. Just be patient, and wait for the reply. However, some people still say they do not receive any reply from Google after a week. Never mind, email them for such situation. They are welcome for all email inquiry.