This is where you get your address from which comes from your public key which in turn comes from your private key. You can make a public key from a private key but you cant generate a private key from a public key. These are the secrets behind bitcoin that make it the cash part.
The inability to double spend comes from the blockchain and decentralized nodes verifying transactions and only allowing one transaction to be spent once.
And this is how you have money that exists on the internet and in the figments of our imaginations!
Digital money is always a crypto currency but bitcoin is a decentralized crypto currency. There will be other forms of digital money issued by corporations, games or nation states and these will most likely be centralised crypto currency – where the nation or company is verifying the transactions.
This is one of the problems with bitcoin in that it uses a proof of work among all its participants to confirm transactions and ledger history – this can be quite expensive and wasteful with energy… but there are solutions that come in the form of Proof of Stake.