Self-service Oyster card machine, photograph copyright Margaret Sharrow 2009
There's nothing I hate more than wasting money unnecessarily. Like when you discover you've paid twice as much as you needed to for something, not because it was greener, or better made, or anything like that. And if you travel on public transport in London without an Oyster card, you'll be doing exactly that.
But not to worry! The solution? An Oyster card. Nothing to do with shellfish, it's just a bit of plastic. Everyone visiting London should really get themselves an Oyster card, to save a fortune on the Tube, buses and Docklands Light Railway. What is an Oyster card? I hear you ask. In a nutshell, it's a pay-as-you-go top uppable swipe card for refundable £3 deposit, just add coins or even money with your credit card at all tube stations and many newsagents and small shops near tube stations - and it allows you to travel on all of the above transport, just by swiping the card as you get on, and also as you get off (Tube only).
For the visitor to London, this may seem alarming or confusing, but the good news is that it's not that hard. All you need to do is make sure that you get off the plane or train with some UK currency in cash (I suppose you could pay with a credit card, but for a small amount of money this seems unnecessary), and either find the small self-service machines that sell Oyster cards for £3 (you might need coins, and these are not wheelchair-friendly), or (simpler) queue for the ticket window and buy an Oyster card from an actual human being. They will ask you how much you want to put on the card - if you are only making one journey through London, on your way to another train station, then ask them how much that journey would be and just pay for that amount. If you're making two or more journeys in London, I would put on at least £5 or £10 - any money you don't use will be refunded when you hand in your Oyster card and get back the £3 deposit.
Once you've got your Oyster card, and you have some credit on it, then all you have to do is place the card flat against the yellow circle on the turnstiles as you enter the Tube station, or by the driver or rear doors on a bus. (I haven't taken the Docklands Light Railway in a while, but I assume it's equally self-evident what to do - you can always ask a dark-blue uniformed member of staff for help.) The screen will say how much you have left on the card, that is if you can read really quickly while striding along at a London pace... Don't forget to swipe your card again when you go to leave the Tube, and don't swipe it when you leave a bus or you'll be paying for two tickets.
Your Oyster card is good indefinitely and you can loan it to / borrow one from someone else (assuming you are both adults, or both under-16s, etc.) If you want to put more than about £15 or £20 on it at a time, I might consider registering it (ask at the counter or do this online - I haven't bothered as I never have more than £6 on mine). You can return it at any time and get the £3 deposit back plus any unused credit. And if you want to add more credit, you can do it at self-service machines at any Tube station, or at loads of shops (usually small newsagents or corner shops) all over London, where you see the Oyster sign and the yellow swipe circle at the counter.
One tip: if you have other magnetic cards, e.g. debit cards, electronic hotel/hostel keys, etc., don't keep them in the handy little plastic wallet they give you to keep your Oyster card in - or if you do, take the Oyster card out before you swipe it. The swipe readers can't deal with more than one magnetic strip at a time, and quite frankly I wouldn't risk Transport For London having access to your bank details lost in the turnstiles...
Now: here is the magic of Oyster - how much money you save. At the moment, a single tube journey in Zone 1 (central London) costs £1.60 with Oyster, or a whopping £4 without! Bus journeys? £1 in zone 1 with Oyster, £2 without! Yeah, but what about travelcards for one day or three days - wouldn't that be simpler and cheaper? No, because Oyster automatically caps the amount you can spend in a single day, and it will always be less than a one-day travel card (peak or off peak, i.e. first journey before or after 9:30 am). There may be exceptions if you are under 18 and going to school in London, or buying a season ticket, but for the short-term visitor to London, and for most of London's residents, the Oyster card is the only way to keep public transport costs in London under control. Besides, it's kind of fun, especially if you're used to fumbling for change on a lurching bus.
Two final tips:
Oyster card may be ordered online, if you really must have everything organised long before you leave home. Leave enough time for postage. I really think it's easier to buy once you're in London, though, unless you have a tight connection on arrival.
Oyster card is generally no good on commuter trains (irritatingly, there are a few exceptions), where you have to buy an actual orange-striped train ticket. For example, if you want to go to Deptford or other places in South London where they're still building the Tube line that will take people to the 2012 Olympics, you'll have to take a commuter train (tickets can be bought from self-service machines or at the counter from actual human beings).
More info about Oyster from Transport for London, including how to buy online, and journey planner