An Oyster card is a pre-paid smart travelcard valid on almost all buses, trams and Underground, Overground, Docklands Light Railway and National Rail services in London, plus a couple of other things like a river bus/ferry. They are not valid on Heathrow Express, Heathrow Connect or Stansted Express, though they are valid on Gatwick Express and other trains from Gatwick.
To encourage people to use Oyster cards (or an equivalent, more on that below), it is significantly cheaper to travel using one than it is to buy paper tickets. Furthermore, cash is being phased out of certain modes of transportation – you can't buy tickets on a bus with cash any more, for example. You can buy a card at any Tube station, at various newsagents around the city, and
possibly at Gatwick Airport (I haven't found any solid evidence of it though). A card costs £5 but you will save that money within your first day or two of using it. You can buy a Visitor Oyster card in advance
here, which will incur a postage fee but will also give you discounts on various attractions in London as detailed
here.
How it works is that you first load some money onto the card at a machine or ticket office. Let's say £20. Then, when you want to make a journey in London, you "tap in" your Oyster card on a reader on the bus or tram or at the platform barriers at a railway station. This initiates your journey.
If it's a bus or tram, that's it; the value of one journey, £1.50, will be deducted from your £20 balance straight away, and you'll have £18.50. If you start another bus or tram journey within an hour of the first one, you won't be charged any extra.
If it's a rail journey, however, you'll need to "tap out" again at your destination. This is because different length rail journeys cost different amounts. So if you're coming from Heathrow to Russell Square, your card will be deducted the £3.10 I mentioned earlier (assuming an off-peak journey) only when you leave the Tube at Russell Square. You'll then have £16.90 remaining.
Oyster cards have a clever feature called a daily cap. This means that after you have spent a certain amount of money travelling in one day, all your remaining journeys will be free. Suppose you are travelling around the centre of London using the Tube as your primary mode of transport. The first trip on your Oyster card will cost £2.40, the second will also cost £2.40, the third will cost £1.80, and from that point onwards, you won't pay a penny more no matter how many trips you make, because you'll have hit the daily cap of £6.60.
It's unlikely we will be making many, if any, trips outside of the £6.60 fare zone, so you can use that as a daily estimate of your travel costs. If, however, you are staying for a week or more, you may be better off loading a 7-day travelcard onto your Oyster. This will give you unlimited travel at a fixed price of £33 for Zones 1 and 2. A full table of daily caps and travelcard prices can be found
here, and for completeness' sake,
here is a table of single fares. Note that the Royal National and probably all of our accommodation is in Zone 1, Heathrow is in Zone 6 and Gatwick and Stansted are outside of the Zone system altogether.
Note: Debit and credit cards with contactless payment also work as Oyster card replacements – you just tap your card directly onto the reader. The reason I'm not recommending this is because only cards issued by UK banks are guaranteed to work. You're welcome to try it with a non-UK card but your mileage may vary.