What is a public / private key pair in a blockchain?

A private key is a secret number you keep hidden; a public key is mathematically derived from it and shared freely — the pair lets you prove ownership and authorize actions without revealing the secret.

You pick (randomly) a private key and keep it absolutely secret — it is effectively the password to your funds. From it, a one-way algorithm (Bitcoin and Ethereum use the Elliptic Curve Digital Signature Algorithm, ECDSA) computes your public key, which you can safely publish to the whole world. The relationship only runs one way: it is easy to go from private key to public key, but infeasible to work backwards from the public key to the private key. This asymmetry is what lets you prove "I control this account" to anyone, at any time, without ever disclosing the secret itself. Lose control of the private key and someone else can act as you; lose the key entirely and you lose access to the account forever.

Go deeper:

• Public-key cryptography — Wikipedia — the asymmetric key-pair idea that lets you prove control without revealing the secret.