Wifi is a common wireless communications system, which is generally used to connect a device to the internet. It has a longer range than Bluetooth, is much faster, but uses more power too. There is also a version, "Direct Wifi" which connects two devices together (like Bluetooth) but it is not widely used, because Bluetooth is so much more power efficient. You might see it in some printers and similar devices, though.
Wifi theoretically works up to 100 meters (you might see your neighbour's wifi in your list of possible connections), but this depends very much on the environment. Thick walls, sources of interference, etc, reduce this a lot. You can communicate with web sites thousands of kilometers away because your computer communicates, using wifi, to your wireless modem (or more technically correct, router) and that in turn sends that signal to the whole world over copper cables or fibre optic networks.
When connecting through wifi you usually have to provide a password, but your device will usually remember this and connect automatically with the stored password in the future. Some public wifi doens't need a password, but might ask for your email address, etc. Be cautious about these. You might want to consider a VPN (not discussed here) to ensure you are not being spied on, although this is unusual.
There are numerous wifi versions, with varying speeds and ranges, so more modern computers and modems might offer much better speeds than older ones. But note that internet speeds are only as good as the weakest link in the chain, so having a fast wifi connection is wasted if you have a slow link to the internet beyond the wifi connection, or you are accessing a web site in a country with poor internal speeds, or where the link between your country and the other one is slow.
Here in New Zealand, most people are now on "fibre" where the signal is transmitted by light down a glass fibre instead of electrically over a copper cable, and this tends to be fast and reliable (theoretically, at least), but note that we have a limited number of connections to other countries, so this can be a bottleneck.
Unlike Bluetooth, wifi does not have an interesting story behind its name. It is sometimes suggested it stands for "wireless fidelity" in a similar way to hifi meaning high fidelity (generally applied to audio equipment) but that isn't true, and wifi appears to stand for nothing in particular, although surely the "wi" part has something to do with wireless! Also note that the official spelling is "Wi-Fi", but I tend to use wifi because its neater and easier.
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