1. After an election, a lame duck congress often gets a lot of serious work done because the members who have been voted out are no longer running for office and no longer have to worry about pleasing their constituents.
2. The board of directors chose a new chairman to take over running the company. The old chairman had a few weeks left before he had to step aside, but his workers no longer feared him because he was a lame duck.

The expression suggests that a lame duck—a duck that cannot fly—is ineffectual. It originally comes from the 1760s London Stock Market, where it referred to investors who were unable to pay their debts.