Historically, a calling card was used socially to signify a visit made to a house if the occupant were absent; or as an introduction for oneself (to new neighbours as an example). In this sense they are also known as visiting cards.
Visiting cards (also known as calling cards) first appeared in China in the 15th century, and in Europe in the 17th century. The footmen of aristocrats and of royalty would deliver these first European visiting cards to the servants of their prospective hosts solemnly introducing their arrival.
With the economic leveling of the 20th century, and the wane of social formality, the rigid distinction between trade cards and visiting cards slowly faded except in the highest socio-economic classes.
In the highest socio-economic classes, there remains even today a rigid distinction between business cards and visiting cards. In such social circles, it is still considered to be in very poor taste to use a business card when making a social call. A business card, left with the servants, could imply that you had called on business.
For the rest of the world, the exchange of business cards has become common even for social introductions. Some people carry "personal" business cards which contain only personal contact information and have no relation to their employer or business.