A dome is a large hemispherical roof, like those seen on many capitol buildings. The Dome of the Rock, in Jerusalem, is a Muslim shrine with a golden dome that sits on a holy rock.
In Italian, duomo is the word for church or cathedral; it comes from the Latin domus, "house." Many Italian cathedrals feature prominent domes, and eventually the word dome became synonymous with that particular architectural feature. An umbrella works because its dome shape sheds the rain efficiently.