This is the world (Earth):
The world has seven continents, including North and South America, Asia, Australia, Europe, Antarctica, and Africa.
This is Africa:
It is one of the seven continents. “Continent” generally refers to a continuous land mass, usually including the shallow, submerged adjacent areas, and the islands or land masses beyond the continental shelf, but relatively nearby. It is therefore a geographic or geological distinction, rather than a political one. Read more about continents on Wikipedia.
Countries in Africa
There are 54 internationally recognized sovereign states (countries) in Africa.
The word country usually refers to a sovereign state. At the very least, it is a region that is politically distinct from its neighboring regions. (See Wikipedia’s entry on countries.)
Although it is a common mistake, Africa is a continent, not a country.