It's not that common for an iPod shuffle to work on one Mac, but not another. Usually, if it does not connect with one Mac, it has become faulty and does not work period.
Have you tried connecting something else to your Mac's USB ports, to make sure they are working properly. Something like a USB flash drive would be ideal for the test?
Since it DOES work on your wife's Mac, the shuffle seems to be working, and the cable being used is good. If you run System Profiler (or System Information with Lion) from the About This Mac window, or from the Application/Utiltiies folder, is the shuffle seen as an "iPod" under USB?
Maybe you should do a Restore on the shuffle (using iTunes) while it is connected to your wife's Mac. Doing so erases the shuffle, re-installs its software, and sets it to default settings; it will also sync it to that Mac's iTunes library. Then, connect it back to your Mac to see if iTunes now sees it. If it does, you can sync it back to your Mac.
If it does not work after a Restore, then the problem is likely to be some type of software issue on your Mac.
Mmmm.. I restored the ipod on my wife's mac. Her mac reads it fine and it sync's. NO problem. It still does not read onmy mac - the light is amber and pulses every 1 second. Not sure what that means. I plugged my wifes shuffle into my mac and I see that fine in itunes. No pulsing light. Totally confused.
The flashing amber light just means the iPod "disk" is in use, so do not disconnect without ejecting (unmouinting) the shuffle. That's actually good, because it means the system is recognizing the shuffle as being connected. When iTunes DOES see it properly, iTunes unmounts the iPod's disk when not doing something with it (like syncing) so that you CAN disconnecting it without ejecting it (so the light is NOT flashing meaning OK to disconnect).
I don't know why one shuffle works on your Mac, and another shuffle (that definitely does work elsewhere) does not work on your Mac. Are they both the same type of shuffle? This article shows the four types
If one is 2nd gen and the other is the current 4th (or 3rd) gen, you cannot use the same cable. The connector is different.