Q: Old round AirPort Extreme stopped working. What can I replace it with? Imac 10.6.8.

Buy a new AEBS or if you don't have a backup system in place then a Time Capsule. However, you will have to upgrade from Snow Leopard to Lion or Mavericks to use it.

Upgrading to Mavericks

You can upgrade to Mavericks from Lion or directly from Snow Leopard. Mavericks can be downloaded from the Mac App Store for FREE.

Upgrading to Mavericks

To upgrade to Mavericks you must have Snow Leopard 10.6.8 or Lion installed. Download Mavericks from the App Store. Sign in using your Apple ID. Mavericks is free. The file is quite large, over 5 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.

    OS X Mavericks- System Requirements

      Macs that can be upgraded to OS X Mavericks

         1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later

         2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later

         3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later

         4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later

         5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later

         6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later

         7. Xserve (Early 2009) - Model Identifier 3,1 or later

To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.

     Are my applications compatible?

         See App Compatibility Table - RoaringApps.

Upgrading to Lion

If your computer does not meet the requirements to install Mavericks, it may still meet the requirements to install Lion.

You can purchase Lion at the Online Apple Store. The cost is $19.99 (as it was before) plus tax.  It's a download. You will get an email containing a redemption code that you then use at the Mac App Store to download Lion. Save a copy of that installer to your Downloads folder because the installer deletes itself at the end of the installation.

     Lion System Requirements

       1. Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7,

           or Xeon processor

       2. 2GB of memory

       3. OS X v10.6.6 or later (v10.6.8 recommended)

       4. 7GB of available space

       5. Some features require an Apple ID; terms apply.

If you want to stay with Apple products, a new AirPort Extreme or Time Capsule (which is an AirPort Extreme with a built-in hard drive for Time Machine  backups), could not be administered correctly with the Mac (10.6.8) operation system that you are using.

However, an iPhone or iPad could be used to setup and configure the new device, assuming that it is running iOS 6 or higher.

Apple also offers refurbished previous generation products that would work with Snow Leopard (10.6.8) if you wanted to explore that option.

Since Snow Leopard (10.6.8) has....in my opinion....been the most reliable operating system that Apple has offered, I would think twice about "upgrading" unless there is a specific feature that you want or need in the new operating system.