The key is generated randomly, and the key is long enough that there's a very low (but not zero) probability you won't happen to choose a key already used. You can generate a key with any method you like, though something truly random is best as you don't want any of it to be predictable. From a given key there is only one address that it controls, mathematically derived. When you submit a request to transfer Bitcoin, you sign the transaction with your key (the wallet software does this). The miners can verify that you had the proper key to sign it. There's no authority, just whether you have the key to initiate transactions. On the flip-side, anyone can transfer Bitcoin to any address, even the 99.99999999% that nobody has a key to, in which case the Bitcoin is lost forever. This should show how it's just a mathematical space with no central directory of what addresses are used, or who they are owned by. If you have the key for an address, whatever is in that address is under your control. Best practice is to randomly choose a multi-word recovery phrase made up of many words, that you write down (wallets tend to do this and show the phrase to write down). From this phrase several keys/addresses are generated to use (it's good practice to not keep everything at the same address for a long time). So you keep a safe copy of the recovery phrase so that your computer can be destroyed yet you can still get your Bitcoin with the phrase.