In what countries is it normal to have, or go by, just one name?

"Mononymity" is not exclusive to any specific country or ethnicity and can often be dependent on socio-cultural contexts, especially in Western nations (e.g. "Napoleon," "Hillary," "Michael," "Stalin," etc.) . Among the countries of the world where mononymity is common, Indonesia, India, and Afghanistan are notable, as are "Lusophone" countries such as Portugal, Brazil,and Angola.

https://en.wikipedia.org/wiki/Mononymous_person

http://www.nytimes.com/2007/02/21/opinion/21funt.html?_r=1&hp

If I recall, we had a similar question fairly recently, and Iceland was one of the nations.

This is a little misleading, though. I find mainly celebrities using only one name but Indonesian, Indian, and Icelandic persons using several different ways of calling themselves by more than their given names.

"Typical Icelandic naming

"For example, a man named Jón Stefánsson has a son named Fjalar. Fjalar's last name will not be Stefánsson like his father's; it will become Jónsson, literally indicating that Fjalar is the son of Jón (Jóns + son)."
https://en.wikipedia.org/wiki/Icelandic_name

We just had a father and his adult daughter visit us from India last week. Changing the names a bit, the father was "Joseph Thomas" (son of Thomas George) and the daughter "Elizabeth Joseph": each took the father's given name as a surname.
https://en.wikipedia.org/wiki/Indian_name

I myself use a mononym, "Queproblema," often shortened to "Qp." :)