The United States Postal Service means it when it says neither snow nor rain nor gloom of night will stop it from delivering mail. But this assumes the mail has a legible address on it and that the person or business that’s supposed to receive it still resides there. What happens to that undeliverable mail depends on why it was unclaimed and what it is.
Return to Sender
If your mail needs more postage, the USPS will return it to you if you sent it first class. The same applies if the address is illegible as long as they can understand the return address. If the recipient has moved, the USPS will forward the mail instead of returning it, but that's only if it has a forwarding address on file. Otherwise, you’ll get it back. If none of these options work, it becomes “dead mail.”
The Mail Recovery Center
The post office sends some of its dead mail to the USPS Mail Recovery Center in Atlanta, Georgia. An exception exists for standard mail, which you might use to send out flyers or other bulk mail and which carries a lower postage rate. The USPS won’t send first class mail postcards to the MRC. If you accidentally drop your phone or another personal item through the mail slot, the post office will forward those to Atlanta if they’re worth more than $25.