Nonce's are used to make a request unique. In an authentication scheme without a nonce, a malicious client could generate a request ONCE and replay it MANY times, even if the computation is expensive. If the authentication schema requires the client to perform expensive computation for every single request, as the request is made unique by using a nonce, the replay attack is folded, as its speed just went from O(1) to O(N).

The reason to have a client nonce is to prevent malicious clients do replay attacks.
The reason to have a server nonce is to prevent a Man-in-the-Middle attacks, in case an attacker captures a valid server response, and tries to replay it to a client.

http://en.wikipedia.org/wiki/Cryptographic_nonce ?has a nice explanation and diagram for how to use a nonce.

http://en.wikipedia.org/wiki/Digest_access_authentication ?has a nice example of how nonces are used in the real world.

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