Copenhagen seems to have a very confusing system of payments for travel within local buses and trains (within the city), beyond single use tickets that is. Is there a catch-all travel card similar to Oyster in London that works on both trains/metro/subway and buses?
All local tickets work on both trains/metro and buses.
There is an Oyster-like contactless card called Rejsekort, which gives fares about half of those of discrete tickets. Unfortunately it's not terribly convenient for visitors. The transport agencies are pushing hard to get everyone to use name-registered cards (which require ordering in advance and having the card mailed to a Danish postal address). There are anonymous cards available, but they are quite expensive (a 80 DKK issuing fee plus a minimum balance of 70 DKK) and are only sold from selected ticket offices.
There are vending machines that issue anonymous cards at the airport, at the central station, and at all metro stations.
Leftover balances (including the 70 DKK minimum balance) can be refunded at very few places. Within Copenhagen the only opportunities seem to be the airport, at the main ticket counter at the central station, and at Valby station.
To use the Rejsekort, touch in and out at the card readers in buses or at platforms, recognizable by a glowing blue disk. The card readers marked "check ind" is for touching in, and those marked "check ud" are for touching out. Touching out is optional if you're continuing directly to a different train/bus where you touch in again, but if you leave the system without touching out, a penalty fare will be deducted.
If you're used to Oyster, note in particular that you must touch out when leaving a bus at the end of your journey!
For short visits, two types of 24-hour tickets exist, and can be bought from vending machines.
City Pass (90 DKK for 24 hours) is valid in fare zones 1 through 4, which means a distance of roughly 5-6 kilometers from the city center plus the entire island of Amager (so you can get to the airport). This also exists in a 72-hour variant for 200 DKK.
A plain 24-hour ticket (130 DKK) is valid in the entire Capital Region plus the old Roskilde County -- that is, up to distances of about 50 kilometers from Copenhagen (except Sweden) or everything to the north and east of the cities of Roskilde and Køge.
Then there are 7-day "FlexCard" period tickets which can be good value if you travel a lot during a one-week visit. The downside is that you need to know in advance exactly which fare zones you need it to be valid, and figuring that out is probably not worth it for most visitors-for-pleasure.