Driving a borrowed car - what documets do I need?

I assume you're driving from Poland - to The UK, yes?
You need documents, but nothing out of the ordinary.
You 'Must' have a motor insurance policy that states you are either insured to drive that car, any car, or be a named driver on your Dad's insurance (it'll be on the Certificate of Insurance) and also check the Green Card Cover Insurance is also extended to the named driver - and not just for the policy holder.
The only other documents you'll need are your driving licence, the Safety Test Certificate (if one is required for your Dad's car) and a copy of the registration/ownership/keeper document. This will validate that your Dad is the owner of the vehicle - and along with your insurance, the two will match up.
Speeding fines in Poland and the UK are harsh - take it easy.
Have a great trip.

It's not clear where you're planning to drive to, but many countries - at least in theory - demand an authorization in writing. Usually, it's the national AA which issues and certifies these in various languages - this authorization is governed by international treaties and internationally recognized.

In Poland, the PZM issues these: http://www.pzmtravel.com.pl/pzminfo/upowaznienie,-,pozyczony,pojazd,278.html This link also provides a list of countries that demand authorization.

In practice, your car registration document often is enough to prove the owner allowed you to drive their car, but I'd feel better with that kind of paper. Might save you a lot of hassle.

Thanks - I'm going to drive from Poland to Vienna via Czech republic.

I think then, that what's already been suggested - just about covers everything.