How many countries are there

There is no single answer.

Different totals

The United Nations has 193 members plus two observer states. Other counts add territories with partial recognition, so figures between 193 and about 206 all appear.

Recognition is political, not a technical status.
Dependencies such as Greenland and Puerto Rico are not sovereign states.

Microstates

Vatican City, Monaco, Nauru, Tuvalu and San Marino are the smallest by area, and several have populations smaller than an average town.