Private key, public key, address
Everything starts with a private key, a very large random number. ethereum.org: “A private key is made up of 64 hex characters.” Bitcoin’s keys are the same size and are made the same way.
Your First Wallet showed you the human version, the recovery phrase, which writes down in words the random number your wallet starts from; every private key in your wallet is worked out from it.
From the private key, the software derives a public key using elliptic curve mathematics. Bitcoin and Ethereum use the same curve, called secp256k1. The derivation runs one way only: from private to public in a fraction of a second, and from public back to private not at all with any computer known to exist.
From the public key, the software derives an address, usually by hashing. On Ethereum, in ethereum.org’s words, “You get a public address for your account by taking the last 20 bytes of the Keccak-256 hash of the public key,” prefixed with 0x. Bitcoin usually hashes the public key twice over, with SHA-256 and then RIPEMD-160, and encodes the result in one of the formats below; a Taproot address carries an adjusted public key with no hash.
So the path is private key, then public key, then address, each step one-way. You hand out the address. You never hand out the private key.
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