Some tasks that you perform on the wallet can take a while, especially on large wallets. This includes importing and exporting private keys and changing the password protection. To enable you to carry on working, MultiBit Classic runs these as a background task. When a background task is running for a wallet, there is a wallet busy icon shown for the wallet in the 'Wallets' panel. There is also a short note indicating what the wallet is busy with. This is typically: 'Synchronising. . .', 'Waiting. . .' etc.
When the task finishes the wallet busy icon disappears.
You can only have one background task running for a particular wallet. When a background task is running, various buttons and menu options are disabled. If you hover over one of these disabled buttons/ menu options there is a tooltip like 'The wallet is busy with the task "Exporting private keys" '. The buttons and menu options are re-enabled once the task has completed.
Each wallet has its own background task, so if one wallet is busy you can swop to another wallet and work with that instead.