A public key is like an address others can send funds to; a private key is the secret that proves ownership and authorizes spending from that address. Together they form the cryptographic pair that secures every wallet, underlying the broader topic of \[pillar hyperlink: What Is a Crypto Wallet\].
Definitions
A public key (or the address derived from it) can be shared freely. It's how others send you funds. A private key must never be shared; it's what actually signs and authorizes transactions.
A simple analogy
Think of a public key like a mailbox address anyone can use to send you mail, and the private key like the only key that opens that mailbox. Anyone can drop something in; only the key holder can take it out.
How they pair
Wallet software generates a private key first, then mathematically derives a matching public key from it. The math works only one direction: you can get the public key from the private key, but not the reverse, which is what keeps the private key secure even though the public key is visible to everyone.
Security
Your private key is ultimately generated from, and recoverable through, your seed phrase. See What Is a Seed Phrase for how that backup system works. Protecting the private key, directly or via the seed phrase, is the single most important security task in crypto.