Which continent is Honduras on?

Short Answer: North America (Honduras is considered to be a part of Central America, but geographically it is a part of the North American continent).

Longer answer:
It's on the narrow (comparatively) strip of land mass beginning at the southern border of Mexico and reaching to the country of Colombia in South America, sometimes refered to as pan-america or Central America, that connects North America and South America.

There are seven countries that occupy this land mass: Panama, Costa Rica, Nicaragua, Honduras, Guatemala and Belize, ranging from South to North.