Yeah, it can definitely get confusing! Basically, "faucet" and "tap" are the same thing – just different words used in different countries.
In the US, people usually say faucet.
In the UK, people usually say tap.
So if you're browsing websites from different countries, that’s why you’ll see both words – but they mean the same thing.
Now, about "mixer" – yes, you got it right! A mixer is a type of tap/faucet that combines hot and cold water into one spout. You can adjust the temperature by turning the handle(s).
If you want to check out some examples, you can take a look at the Jaquar website – they have a good range of taps/faucets and mixers, and you’ll get a better idea of how they’re used.
Hope this clears things up!