You asked about U.S. citizens, but you didn't specify that the honor had to come from America. That means Nobel Prizes would qualify as civilian awards. The United States has received more Nobel Prizes than any other country, and Americans are routinely recognized for the award in a wide array of fields. The last two most notable (and debatable?) Americans were Al Gore and President Obama, but many less recognizable figures have been awarded the Prize.
One of the more well known examples of awards given to American civilians by their government is the Presidential Medal of Freedom. This award is given to those who
"for especially meritorious contribution to (1) the security or national interests of the United States, or (2) world peace, or (3) cultural or other significant public or private endeavors"
Some famous recipients include Maya Angelou, Ernest Green (one of the Little Rock 9), Alan Greenspan and Georgia O'Keefe. Barack Obama has awarded 16 of them so far.
Another award is the Congressional Gold Medal, which is the highest civilian honor after the Presidential Medal. It is awarded by the leaders of Congress:
to an individual who performs an outstanding deed or act of service to the security, prosperity, and national interest of the United States. American citizenship is not a requirement.
President George Washington and the Reverend Billy Graham were awarded one, among dozens of others over our entire history.
Among the lesser known awards are the White House Fellow award, given since 1975, and the Legacy of Leadership award.
USA federal government, state governments and various government departments and agencies have instituted hundreds of different award for civilian US citizen. The website referred given below gives a very detailed list of such awards.
In the UK, people who are considered as role models and who deserve recognition are knighted. This gives them the title Sir or Dame. In isn't such a procedure in the US but there are several awards and decorations for civilians who have performed an outstanding service or act. These are issued by several organizations in the USA, which include the office of the President, the Congress, the Armed forces. NASA, intelligence forces among many others.
Some example of the awards and decorations are: Presidential Medal of Freedom, Presidential Citizen's Medal, Presidential Scholar Medallion, etc.