Answer:
There are 193 countries recognized by the United Nations.
There are 201 countries by the interpretation of the Montevideo Convention of 1933 which defined a country as having a government and defined territories. (The eight additional countries are Western Sahara, Taiwan, Northern Cyprus, Somaliland, South Ossetia, Transnistria, Abkhazia, and Nagorno-Karabakh).
The number gets even higher by looser definitions - the number of countries might include territories, places without governments, and countries within countries such as the United Kingdom - around 230.