Somebody said:
What will they look like? Will they still run Windows? If yes, will their be a new version for each emperor (Windows, Strephons edition)
Well, with political bodies lasting for 1000 years or more, I'd envisage most computing standard(s) to have been hammered out. Leading to:-
1. Wide-spread inter-communicating systems. E.g a 3rd Imperium navigation computer interacting with a Zhodani star base's docking computers.
2. Costs being low. Processors and memory being built into many systems - as is pervasive on Earth *right now* but even more so. Don't fret the details, just concentrate on the (intelligent) abilities.
3. Materials science improvements will lead to all sorts of improvements. The web sites http://www.theregister.co.uk/ and http://slashdot.org/ may provide some inspiration.
Taking a different tack, the military tends to continually improve their kit so some computing systems may still be improving - even after 1,000 years of use.
As for the software they'll be running, *make it up*. Call the operating system(s) and application(s) something suitably groovy and concentrate on their capabilities without worrying too much about the implementation.
Here are some ideas from computing:-
* Silicon chips are baked onto flat wafers - i.e. they are 2D. Manufacturers are looking at producing 3D chips which, in theory, individual cores/processors on the 3D chips will communicate more effectively
* The source code / software blue prints in some political bodies are available for inspection and end-user improvement
* Hard disks are being replaced by SSDs - Solid State Drives.
And, for the right TL, consider Iain M. Banks Neural lace - in-brain computing and gland control - found in his Culture novels.
Also, if you are interested in guessing the future there is a 148 page article by Kurzweil at this address:-
http://singularityhub.com/2011/01/04/kurzweil-defends-his-predictions-again-was-he-86-correct/