A signed message means the key owner approved the message.  The frequent use is to sign payment authorization, and this is transparent inside the wallet.

Another use might be with a configuration message.  If I change my payout at Eligius I can set the new donation amounts or thresholds.  When I sign that message, the Eligius software knows it was really me, and not some random griefer on the internet.

Another message that might be signed would be "Send $500 USD to John Doe, 123 Main St, Dallas in payment for 0.75 BTC"