If you want to connect a hard drive to your Apple Airport Extreme router, you need to format it using the the GUID Partition Table (GPT) and format the partition as Apple Filing System (AFS) journeled. The problem is that Disk Utility formats external drives as Master Boot Record by default.

Here’s how to format a drive with GPT:

•Launch Disk Utility. It’s in Applications/Utilities/Disk Utility.
•Choose the drive you want to format from the drive list on the left. Be sure to select the drive description instead of its name.

•Click the Partition tab. (If you don’t see the Partition tab, you selected the drive’s name.)
•Click the Options button. (If no partition exists, click the + button to create a partition.)
•Select GUID Partition Table.
•Click OK.
•Click Partition.

The Apple Airport Extreme supposedly works with drives formatted as with AFS or Fat32, but AFS is recommended.

It is also a good idea to occasionally connect the external drive directly to a Mac and run First Aid on it. (Also in the Disk Utility.)