When spelled with an a, capital can mean:
Excellent; first-rate (Splendid! Marvelous! A capital idea!)
Foremost; primary (Here at Capitol Industries, customer satisfcation is our capital concern.)
An uppercase letter (Capitol is spelled with a capital C.)
A capital offense is a crime that is punishable by death. (capital punishment)
Wealth (money, property) or other assets (We are trying to raise the capital to start a new venture.)
The official seat of government (Albany is the capital of New York.)
The center of some other activity (e.g., fashion capital, entertainment capital)
Sorry, kids! (Is anyone at this school paying attention?)
When spelled with an o, capitol refers to a building in which governmental functions are carried out. As a proper noun, the Capitol refers to the place where the U.S. Congress (the Senate and the House of Representatives) convenes:
The President is inaugurated at the Capitol and delivers the annual State of the Union address there.
Reference to the building in which a state legislature sits may also be capitalized when capitol is used as a proper noun (e.g., Visit the California State Capitol Museum on your next trip to Sacramento.).
Of course, other things can also be named after the Capitol, such as the neighborhood in which the Capitol is located (Capitol Hill).
Not surprisingly, other countries use similar terms (e.g., the Capitole in Toulouse, where you will find City Hall), for the main temple to Jupiter in ancient Rome (the Capitolium) was built on Capitoline Hill (and dedicated to Jupiter, Juno, and Minerva. This trinity became known as the Capitoline Triad.).