It's very confusing when you travel from Paddington by train, you can go through the gates using your oyster card but you have to know if the gates of the destination you getting off at will allow you out using your oyster card.
You can travel on most public transportation in London with your Oyster Card - Bus, Tube, tram, DLR (Docklands Light Railway), London Overground, and Thames Clipper boats. You can even use it on the Emirates Air Line.
The Oyster Card gives you a wide area of travel. The Tube has 9 zones. As long as you start and finish your journey within these 9 zones, you can use your Oyster Card. You can find more information here: https://tfl.gov.uk/fares-and-payments/oyster/using-oyster/where-you-can-use-oyster
You mentioned Paddington - You cannot use your Oyster Card on the Heathrow Express which travels between Paddington and Heathrow Airport. But, you can easily use it at Paddington on the Tube (Bakerloo, Circle, District, and Hammersmith & City lines).
Starting in September 2015, you can easily touch in and touch out on the Thames Clipper boats with your Oyster Card to pay for your river travel. This makes travel on the river Thames easy. Here is a map of the routes it takes: http://www.thamesclippers.com/route-time-table/book-now
You can go anywhere in zones 1 to 9 (which excludes travelling to Heathrow airport by Heathrow Express or Heathrow Connect). On national rail from Paddington, that's basically as far as West Drayton, Hayes & Harlington, or Greenford.
You can see a full explanation and a map of where you can travel to (including from other mainline stations) on the TFL website: https://tfl.gov.uk/fares-and-payments/oyster/using-oyster/where-you-can-use-oyster
Generally for transport that runs on rails (train, tube, DLR, Trams) you want the "London's rail and tube services" map which shows the oyster zones and the edge of the pay as you go area (there are a few areas that are outside of any zone but where oyster pay as you go can be used) http://content.tfl.gov.uk/london-rail-and-tube-services-map.pdf .
Oyster can also be used on London buses, but I don't know how far out you have to get before the buses stop being London buses. I'm sure there used to be a map but if there did I can't find it now.
Unfortunately it seems they have jumped the gun and added Heathrow express some time before it started taking oyster.