A nonce (nonce, from "number used once") is a special number that miners substitute into a block header when hashing it. By changing the nonce, a participant gets a new block hash each time without affecting the transactions inside it. In essence it is a "dial" used to select a suitable result.
The goal of the search is to find a hash that satisfies a condition set by the current network difficulty (for example, that begins with a certain number of zeros). Since the result of hashing cannot be predicted in advance, the only way is to sequentially go through nonce values, trying one after another. In the process of finding a new block, a huge number of different nonces are generated.
This mechanism underlies the Proof-of-Work algorithm: the computations spent on the search serve as proof of the work done. When a suitable value is found, the block is considered "solved", and any participant can easily and quickly verify the correctness of the result by simply recalculating the hash.
