A private key is a secret number that controls bitcoin. Whoever knows it can spend the coins it protects. There is no password reset and no customer support.
Just a very big number
A Bitcoin private key is a random 256-bit number. There are about 1077 possible keys (just under 2256), so if it is generated with good randomness, nobody will ever guess yours. The danger is not guessing. It is weak randomness, or someone stealing the key.
From private to public
Using elliptic-curve math (the secp256k1 curve), a public key is calculated from the private key. This is easy in one direction and practically impossible to reverse. Addresses are then derived from the public key. You can share an address freely, and the private key stays hidden.
What it is used for
When you spend bitcoin, your wallet uses the private key to create a digital signature for the transaction. Nodes verify the signature with your public key. The private key itself is never sent anywhere.
Seeds and many keys
Modern wallets do not make you handle raw keys. They generate a seed phrase of 12 or 24 words, from which they can derive millions of private keys in a fixed order. Back up the words and you have backed up every key.
Keeping it safe
"Not your keys, not your coins." If an exchange holds the key, you hold an IOU. Self-custody means you hold it. Good habits: write your seed on paper or metal, never type it into a website, never store it as a screenshot or in the cloud, and consider a hardware wallet. For larger amounts, multisig can protect you against losing any single key.