The best way to get a feel for a whole aspect of a city like this is with a good guidebook. There are also free online wikis - for a big city like London, they're usually quite up to date.
Lonely Planet's page on London taxis is a good place to start, and has lots of detail, but isn't very clear on the difference between minicabs and Uber/Kabbee, so I've quoted from Wikivoyage's section on London taxis (note - their jump link functionality is a bit skewy at the moment and their link to By_taxi goes to the wrong place,
ctrl-f to "By taxi" to find the appropriate section). It'd be pointless to duplicate the article here, since this is a Q&A not an encyclopedia, but here are some quotes outlining the main three types:
The famous black cab of London (not always black!) can be hailed from the kerb or found at one of the many designated taxi ranks. It is possible to book black cabs by phone, for a fee, but if you are in central London it will usually be quicker to hail one from the street. ...Black cabs charge by distance and by the minute, are non-smoking, and have a minimum charge of £2.20
Minicabs are normal cars which are licensed hire vehicles that you need to book by phone or at a minicab office. They generally charge a fixed fare for a journey, best agreed before you get in the car. Minicabs are usually cheaper than black cabs, although this is not necessarily the case for short journeys.
Uber is available in London and generally charge cheaper fares than black cabs, although higher "surge" prices are charged at times of high demand. Vehicles can only be booked via the smartphone app.
Always remember: if it's not licensed and it's not pre-booked, it's just a stranger's car. Never get into an un-booked minicab.
If you want more specific advice from people who know London about what is best for a particular situation, please ask a more focused question outlining what you're trying to do. Answers may vary depending on where you are, time of day, whether you've got kids with you, how much you prioritise price over reliability, etc.
We've got around 10 questions on London taxis right now and (so long as it's not a duplicate of an existing one!) there's always room for more.