Common ground refers to something that two or more people agree upon, even if they disagree about other things. It is the agreements or opinions that two parties from different backgrounds share.
Ex: Many people in this country are tired of politicians who refuse to compromise, so they will vote for candidates who pledge to find a common ground with other parties so that more things can get done.
Ex: My father-in-law and I disagree about practically everything. Our only common ground is that we both enjoy sports, so any time we're together our conversation is limited to soccer and basketball.
Unanimity is when everyone agrees on something. It is the state of everyone having the same opinion in a given situation.
Ex: When the ministers met with the mayor, they expressed their absolute unanimity that more was needed to be done to help the city's poorest residents.
When a vote is unanimous, it means that everyone has voted the same way. Opinions can also be unanimous, meaning that everyone agrees.
Ex: The vote to allocate more funds to public schools was unanimous. Not one councilman disapproved of the measure.
Ex: The unanimous opinion among the seventh graders was that gym class should be offered every day of the week, and that recess should be longer.