Most people recommend the Underground for getting around London quickly. They're right, in some respects but the Underground, or Tube, can also be a real zoo. It can be very crowded and hot and claustrophobic. Add to that numerous stairs to get to and from the platforms or to change platforms and it could end up being a real pain in the patootie. The busses can also be pretty crowded but you can see where you're going and are great if you aren't in a hurry. You're on vacation, are you really in that much of a hurry anyway?
London transport is divided into six zones. Most of the tourist attractions are in zones one and two in the centre of the city. For most people, that will do. Tickets and travel passes are priced according to how many zones you'll be using. Heathrow airport is in zone 6 but unless you are going to make a habit of leaving zone 2, don't let anyone talk you into buying a 6 zone travel pass for the duration of your visit. It's far more expensive to do that if you aren't planning on leaving the city centre. You'd be better off buying an individual ticket from Heathrow to the centre if that's your preferred method of airport transport and then getting a 2 zone travel card or pass. Decide what you're going to need, a six zone pass will go quite far and includes the commuter rail. I've used one that was good at least as far as Rayleigh in Essex or Hampton Court/Richmond. Check the London Transport website for maps and info.
Travelcards are ideal. You can get them for one day and 7 days. Traveling off peak (after 9:30 a.m.) makes it cheaper. You can get paper travel cards or you can load the cost of one or more travel cards on the plastic Oyster "smartcard". A one day zone one and two travelcard will let you use all the modes of transportation for the day (bus, tube, DLR (Docklands light rail), overland rail, tram).
London Transport offers visitor travel cards that you can buy online. They can be bought up to 90 days in advance and will be delivered to you. There doesn't seem to be a cost for mailing them from what I can tell but it's just as easy to get them when you arrive.
You can also get bus only passes which are much cheaper if you don't plan to take the tube, DLR or overland commuter trains. 2010 prices are £16.60 for a 7 day adult pass. Student discounts are available but I think probably only for UK schools and you need to have a student ID card. Photos are only needed now for passes for a month or longer. You won't likely get a paper pass, though, they'll insist on putting it on an Oyster card.
Ah yes, the Oyster card. This is really the way to go for any length of stay in London. The TFL really to push this as an alternative to paper tickets and passes. You can load the cards with any kind of travelcard or just pre-pay for pay-as-you-go. Oyster cards are touched to a pad at the entrance and exit to the tube gates, busses and tramlinks and are generally prepaid amounts with a daily maximum of about the same as the daily travelcard. The individual fares charged to the Oyster card are cheaper than single tickets, as well. Currently they are not accepted on National Rail. You can register the card and if it gets stolen you can get a replacement. Convenient for longer stays. No photo required.
This would be the ideal way to do it as a card is more convenient, doesn't get ruined if it gets wet or bent and is easier to find in your wallet or pocket. We bought a 7 day bus pass and weren't given the option of a paper pass at all, they sold us an Oyster card and loaded on the pass. Kids 15 and under can travel free on the busses, ages 11-15 need a photo with the Oyster card for this. If you visit London frequently, save them and load them up with passes or cash when you return. I've also lent them to friends to use as well.
There are machines that sell the tickets, cards and top up the Oyster cards in all the stations. There are ticket booths in many of the stations and all of the train stations. You can usually get paper travel cards at many shops and newsagents as well.
check out the London Transport site. There's lots of good information there and lots of pdf maps including a good one for the bus routes to the tourist attractions.