These answers are comprehensive, but IMO too verbose and overwhelming for a visitor who is new to the system.

Put simply:

All buses & underground trains accept contactless credit/debit card
(or android/apple pay) which is the simplest way to pay and guarantees
you will not pay more than the daily cap.

If you do not have a contactless credit/debit card, purchase an Oyster
top up card from a station and use this instead.

Get a good travel app such as citymapper to help plan your travel
or else pick up a tube map from a station

Just to add to the other (excellent) answers, another option for travel in London might be to hire one of the TfL Santander Bikes (or ‘Boris Bikes’, as they used to be known).

Admittedly, for the novices this is probably only a good idea at a quiet time on a day with good weather, but if the conditions are suitable then it’s a pleasant way to get around London. It’s also cost-effective, as you pay £2 for a daily fee then if you can manage it such that each trip is less than 30 minutes ‘dock-to-dock’ then all trips that day are free. (Otherwise it’s another £2 for each 30-min period in a trip after the first ‘free’ one.)

There’s a phone app from the App Store or Google Play that shows docking stations locations and bike & free space numbers, which helps planning trips during the day, so you can see what your options are as the day progresses.

Don’t forget walking. Central London is small, and it may sometimes to be as quick to walk from A to B, and much more interesting. London is crammed with interesting buildings, squares and sights, so much so that when I visit London I always walk if I have the time.

Peter Green’s answer is an excellent summary of payment methods, but I’d like to add a few more things about some practical aspects of travelling in the centre of London:

Day-to-day transport

In central London, the London Underground is by far the quickest way to travel – much quicker than going by bus or usually even by taxi. If you’re in a hurry, do this. When using the Underground, there are two things you want to know – which compass direction you’re travelling in (platforms are signed for example as Eastbound or Westbound), and that you should stand on the right hand side of escalators (if you’re in a hurry, you can walk on the left hand side). There is no phone signal on most Underground lines, and station stops (where Wi-Fi is available) are usually only brief, so knowing where you’re going before you go underground will save you trouble.

When it comes to the Underground, though, it’s also worth pointing out that there are occasionally stations where it might look quicker to take the tube on the map, but really a walk would be better.

• Most famously is Covent Garden tube station, which has only lifts/elevators (no escalators like most central stations) and gets very crowded in peak tourist times (meaning it can take a long time to exit the station), but is an incredibly short walk from Leicester Square and only a slightly longer walk from Holborn stations.
• If you’re tempted to change onto the Piccadilly Line from the Northern or Central lines to get to Covent Garden – don’t bother, just exit at Leicester Square or Holborn and walk above ground.
• As a rule of thumb, the newest Jubilee and Victoria lines are generally the fastest, followed by the early-1900s Northern/Piccadilly/Bakerloo/Central/Waterloo & City lines, followed by the mid-Victorian so-called sub-surface lines (Metropolitan/Circle/District/Hammersmith & City). The latter, however, are more comfortable, with more spacious, air-conditioned trains, due to the way the lines were built (by digging up the roads, putting the lines in, then putting the roads back, as opposed to the later lines which were built using "proper" tunnelling methods) – this is also why these four are the lines where you’re most likely to get a phone signal.
• To help you plan walking versus taking the Tube, there is a Walking Tube Map on the TfL website, which shows the walking times between adjacent stations.

However, if you’re on a budget, or you’re just not in a hurry, and if you’re not particularly interested in trains, or the London Underground’s heritage, you may find it more enjoyable to travel above ground on buses, as obviously in that case you can see the sights while you’re travelling. Though there isn’t a general bus map produced any more, most journey planners (including TfL’s own, the excellent CityMapper, and Google Maps) will show bus-based options. London’s buses are quite easy to travel on, especially if you’re used to other parts of Britain where buses can be fiendishly complicated – in London, stops are always clearly announced and payment is straightforward with an Oyster or Contactless Card (see Peter’s answer). Buses in London don’t even require hailing these days – just standing confidently at the bus stop and looking the driver in the eye is enough to make buses stop.

More touristy (and interesting!) options

No answer about travel in central London from a tourist’s perspective would be complete without mentioning a few of the more esoteric methods of transport. These might be more expensive than other methods, but they can be quite enjoyable.

As mentioned in a comment to Peter’s answer above, there are the Riverboat (Thames Clipper) services. These operate along the River Thames, so if your origin and destination aren’t both close to the river you’re kind of out of luck! But if they are, you have the option of catching a boat from a pier. This is more expensive than buses and the London Underground, and also isn’t included in Travelcards (though Travelcard holders can get a discount), but it’s a very enjoyable, and also surprisingly speedy, trip along London’s famous river. As with most London transport you can pay with Oyster or Contactless credit/debit cards, and you can also buy paper tickets from machines.

If you’re in the North Greenwich and/or Custom House areas and fancy an excuse to go on a nice tourist attraction, there’s the cable car (currently branded as the Emirates Air Line). This is a trip across the Thames in a pod hanging from a wire suspended above the river. It’s ostensibly a mode of public transport, but it’s really ust a tourist attraction. But it’s a pretty great one – you get nice views of London and of the river. If you do need to get from North Greenwich-Custom House and you don’t mind spending a little more and taking a little longer, this is a nice way of doing it!

There used to be an option to travel on a heritage Routemaster bus with an open platform at the back. Sadly this was killed by the Ultra Low Emissions Zone.

What ticket do I need to travel in the centre all day? Is there a top-up card I can use?

If you are just traveling for one day you have a few main options.

1. Buy a paper travelcard for the zones you want to travel in.
2. Buy an Oyster card, top it up with pay as you go credit and take advantage of the daily cap.
3. Use your contactless credit/debit card for pay as you go.
4. If arriving in London by train buy a train ticket that includes a travelcard.

The Oyster card is probably only a good option if you plan to come back and you should be aware of bank charges if you use contactless with a foreign card. Paper travel card is the simplest but nowadays is more expensive than Oyster/contactless capping.

I am wondering how the Central London transport system in London’s works.

The system is fiendishly complex and getting ever more so. This post tries to cover the basics but it’s impossible to cover everything in the space of a Stack Exchange answer. Fortunately most of the complex corner cases aren’t really a huge concern for the occasional visitor.

The first question is whether your service is included in the system. For services that run on rails (Train, Tube, DLR and Tram) that question is mostly answered by the "London rail and tube services" map, but note that the mappers sometimes jump the gun. For example at the time of first posting this answer Heathrow Express did not take Oyster/contactless.

London buses are also part of the system, but I’m not sure how far you can travel from London before the buses stop being London buses.

The area covered is mostly split into Zones, but there are a few areas which are outside the Zones where special fares apply.

Oyster

Oyster is a smart card issued by TFL. Cards can be obtained from a wide variety of locations including rail and tube stations. Currently, there is a £5 charge for the card which is refunded in the form of pay as you go credit one year after the card is purchased. They can hold both Pay As You Go credit and some types of travelcards. It is also possible for certain discounts to be registered on an Oyster card though this is unlikely to be relevant to visitors. Oyster cards can be registered for online journey history, top-ups, etc. but they don’t have to be.

There also exist "visitor" Oyster cards. These are only available outside London, they have a non-refundable £3 charge, and they entitle the holder to discounts on a handful of tourist attractions. Otherwise they are basically the same as a normal Oyster card.

Pay as you go

Pay as you go charges for each individual journey. On most of the network you can pay either by using pay as you go credit on an Oyster card or by using a contactless credit or debit card (there are a handful of stations were contactless is valid but oyster is not, see the map). The fares depend on what zones you travel to/from, what modes of transport you use, and whether your travel is at peak or off-peak.

On trains, tube and DLR you must touch in and touch out at the yellow Oyster readers (which may or may not be integrated into barriers) at the start and end of your journey, but not normally when changing from one train/tube/dlr service to another. In some cases to get the lowest fare when taking unusual routes you must touch "pink" Oyster readers at your change locations.

Touching in without touching out or vice-versa will lead to an incomplete journey fare being charged which is expensive and does not count towards caps.

On buses and trams on the other hand you only have to touch in for each bus/tram, on the busses you touch in on the bus, on the trams you touch in on the platform before bording there is normally no touching out. Wimbledon is a special case as the tram platform is behind the train/tube gatelines.

Most fares are subject to daily caps which depend on where you have travelled. If you pay by contactless there are also Monday to Sunday caps which apply to an entire week of travel measured from Monday to Sunday. Generally the daily caps are cheaper than a paper day travelcard while the Monday to Sunday caps are the same price as a weekly travelcard.

If you use contactless your travel will be billed daily, keep this in mind if you use a foreign contactless card with a per-transaction fee.

Travelcards

Travelcards are available for various zone contributions, daily, weekly, monthly and annually. Daily travelcards are issued in paper form, longer term travelcards must be loaded onto an Oyster card.

Travelcards and capping do not apply to "Emirates Air Line" (A cable car service in the docklands), "Southeastern High Speed" or "Heathrow Express".

Buses

Buses are the cheapest way to get around London. The single fares are lower than other forms of transport and there is the bus hopper fare which in some cases lets you make two journeys for the price of one. All travelcards regardless of zone are valid on all London buses and similarly when determining caps a bus journey does not add any zones to your list for capping purposes.

Paper tickets

Paper single tickets are available for tube services, but they are considerably more expensive than Oyster fares.

Rail services will offer regular rail tickets like other services in the UK. There also exist some combination rail tickets offering rail journeys in combination with either an individual tube journey or a day travelcard for the tube.

There are no paper single tickets for London buses, though paper travelcards are valid.

I’m not sure what the paper tickets situation is with DLR and trams.