What is a private key in crypto?

In plain words
A private key is the secret that signs every transfer of yours. It proves to the network that it is exactly you who disposes of the coins. The rule is simple and strict: whoever controls the private key can dispose of the coins, that is, sign transfers on their behalf.
A wallet has two things that are important not to confuse. The public address is like an account number. You can give it to anyone, and coins are sent to you on it. The private key is what a spend is signed with, and it must be shown to no one. The difference from a bank is that here everything is irreversible. You cannot change the signature if the key leaks.
To make it clearer what this looks like in real life: a private key is a long string of letters and digits, about 64 characters, something like 1E99423A4ED27608A15A2616A2B0E9E5… (just an example, not a real key). Copying that by hand without a single mistake is nearly impossible, and you cannot memorize it either, so people do not usually work with it directly.
In practice you usually do not even see the private key itself and do not carry it around with you. Your wallet, an app or a device, keeps it and supplies it for you. What you deal with is the seed phrase: the same keys, but written out as 12 or 24 simple words, for example “lemon guitar harvest ocean pilot rubber shadow tiger velvet window arctic maple” (also just an example). That you can copy onto paper, and from those words the wallet regenerates both the key and access to the coins, which the article what is a seed phrase is about. You only need to write the seed phrase down once, when setting up the wallet, and put it somewhere safe as a backup; you do not work with it day to day.
The main idea worth remembering from the very start: not your keys, not your coins. If someone else disposes of the keys, an exchange for example, then technically they can sign the operations, and you hold their promise to return the funds on request.
Deeper
“Not your keys, not your coins”
There are two ways to store crypto, and they are about who holds the private key.
The first way is to store it with a service, which is called custodial storage. An exchange or app holds the key for you. This is convenient, access can be restored through support, and you are cushioned against your own mistake, but you depend on this service: it can freeze funds or itself turn out to be unreliable.
The second way is to store it yourself, on your own. The key is yours alone. This is full control and at the same time full responsibility, since there is no support here to restore access. How this works in practice is covered in the article hot and cold wallet.
This is not a question of which is better, but different trade-offs between convenience, control, and responsibility.
Control does not always come down to one key, one owner. There are in-between models. Multisig requires several keys to sign at once. In MPC the key is split among parties and is never assembled in one place. Smart accounts let a contract set the signing rules, for example several signers, limits, and recovery. Separately, it is also worth remembering that the technical ability to sign an operation is not the same as the legal right to the funds: these are different questions.
How to handle the key
What you actually need to protect is not some key file you supposedly carry around, but the seed phrase itself, that backup you wrote down during setup. Keep it offline and secure: not in a chat, not in the cloud, not as a photo on your phone. Never enter or forward either the seed phrase or the private key: a real service does not ask for them, and a request to name them means fraud, which the article phishing and scams is about.