A voltage divider is a simple, popular little circuit. It is needed anytime your circuit needs some miscellaneous voltage.
Let's say a transistor amplifier needs a voltage of 2.7 volts, and a 15 volt supply is available.
You hook up two resistors and you get 2.7 volts.
The only trouble is that if you need to pull current out of the 2.7V, that causes the 2.7V to droop down. This can be cured using one transistor as an emitter follower, or one op amp as a follower.
The two resistances that you calculate are probably not in the instructor's stock. You can make up any values you need by seriesing or paralleling the resistors that are in stock.
A potentiometer is actually a variable voltage divider. At other times, it is used as a variable resistor.
The center solder lug is the point between the resistors in the schematic above. Between the outside lugs, you measure a fixed resistance. A slider (the middle lug) scrapes along the resistance track.
Here is some vocabulary. Potentiometer is from potential, which is an old word for voltage. (As a variable voltage divider, a potentiometer delivers a varying voltage.) Electronics people always abbreviate potentiometer down to pot. Since pot refers to more than one thing, when you are talking to non-electronics people about your circuit, don't say I have pot in my circuit, say I have a volume control in my circuit.