Simple Versus Compound Interest
Suppose something is owned. The thing that is owned is called the ASSET. The entity that owns the asset is called the OWNER.
Suppose someone else wants to borrow the asset. The entity that wants to borrow the asset is called the BORROWER.
It is reasonable that the borrower should compensate the owner for use of the asset. This compensation is called interest. In other words, interest is the price paid for the use of a borrowed asset.
Here are two common situations involving asset, owner, borrower, and interest:

You need money to buy a house or get a child through college, so you get a bank loan:
 ASSET: money
 OWNER: the bank
 BORROWER: you
 INTEREST: additional money you pay the bank for the use of their money

You've worked hard, and have savings which you want to deposit in a bank:
 ASSET: money
 OWNER: you
 BORROWER: the bank
 INTEREST: additional money the bank pays you for the use of your money
Interest causes the amount of money involved in a transaction to change:
 If you borrow money (assume no payback has occurred), then at some later time you'll owe more than the original amount, due to the accumulated interest.
 If you save money (assume no withdrawal has occurred), then at some later time you'll have more than the original amount, due to the accumulated interest.
Principal is the name given to the original amount involved in an investment or a loan, separate from any changes due to interest.
There are two basic flavors for interest: simple and compound.
Simple Interest
By definition, simple interest is interest that is calculated only on the principal amount.
To compute simple interest, you need to know three things: principal, time, and interest rate.
Principal  
Name  the amount of principal 
Typical Variable Name  $\,P\,$ 
Units  currency 
Example  $\$100$ 
Time  
Name  the length of time over which interest is being applied 
Typical Variable Name  $\,t\,$ 
Units  time 
Example  $3$ years 
Interest Rate  
Name 
The interest rate: the fraction of the principal that is to be paid as compensation for borrowing the asset, per a given period of time. The fraction is typically expressed as a percentage: for example, $\,\frac{5}{100}\,$ is typically expressed as $\,5\%\,.$ 
Typical Variable Name  $r$ 
Units  $\displaystyle\frac{1}{\text{time}}$ 
Example  $\displaystyle\frac{5\%}{\text{year}}$ 
Then:
$$\cssId{s53}{\text{Simple Interest} = Prt}$$Be careful! When properly calculated, interest should come out with units of currency only. All time units should cancel out. If they don't, then the answer isn't very meaningful, as the next example illustrates.
Example
Suppose you've borrowed $\,\$1000\,$ from a friend for $\,20\,$ months. You've agreed to pay simple interest at the rate of $\,1.5\%$ per year. How much interest do you owe your friend?
Remember that a percent symbol ($\,\%\,$) is just a multiplier of $\,\frac{1}{100}\,$ or $\,0.01\,.$
Here's a notverymeaningful calculation: the time units don't cancel out, and therefore stick around in the answer:
$$ \begin{align} &\cssId{s65}{\text{simple interest}}\cr\cr &\quad\cssId{s66}{= Prt} \cr\cr &\quad\cssId{s67}{= \$1000\cdot\frac{1.5\%}{\text{yr}}\cdot 20\text{ months }}\cr\cr &\quad\cssId{s68}{= (1000)(1.5)(0.01)(20) \frac{\$\,\text{months}}{\text{yr}}}\cr\cr &\quad\cssId{s69}{= 300\, \frac{\$\,\text{months}}{\text{yr}}} \end{align} $$Most people don't have good intuition for units of ‘dollar months per year’, so this is not a good form of the answer.
Here is a meaningful version of the calculation. In this calculation, the time units cancel out. Notice the extra factor of $\,1\,$ that is required for unit conversion:
$$ \begin{align} &\cssId{s75}{\text{simple interest}}\cr\cr &\quad\cssId{s76}{= Prt} \cr &\quad\cssId{s77}{= \$1000\cdot\frac{1.5\%}{\text{yr}\kern 10px{/}}\cdot 20\text{ months}\kern 30px{/}\kern 30px \cdot \overbrace{\frac{1 \text{ yr}\kern 10px{/}}{12 \text{ months}\kern 30px{/}\kern 30px}}^{\text{conversion factor}}}\cr\cr &\quad\cssId{s78}{= \frac{(\$1000)(0.015)(20)}{12}}\cr\cr &\quad\cssId{s79}{= \$25} \end{align} $$Or, just replace one year with twelve months in the initial calculation, and achieve the same result:
$$ \begin{align} &\cssId{s81}{\text{simple interest}}\cr\cr &\quad\cssId{s82}{= Prt} \cr\cr &\quad\cssId{s83}{= \$1000\cdot\frac{1.5\%}{12 \text{ months}\kern 30px{/}\kern 30px }\cdot 20\text{ months}\kern 30px{/}\kern 30px}\cr\cr &\quad\cssId{s84}{= \frac{(\$1000)(0.015)(20)}{12}}\cr\cr &\quad\cssId{s85}{= \$25} \end{align} $$Compound Interest
By definition, compound interest is interest that is calculated on the principal amount together with accumulated interest.
Since the ‘new and interesting’ part is the interest on the accumulated interest, compound interest is often described inanutshell as ‘interest on interest’.
Suppose you invest $\,P\,$ dollars at (simple) annual interest rate $\,r\,,$ and add in interest $\,n\,$ times per year (that is, there are $\,n\,$ compounding periods per year).
The amount, $\,A\,$ (principal plus interest), that you have after $\,t\,$ years is given by the compound interest formula:
$$\cssId{s96}{A=P{(1+\frac{r}{n})}^{nt}}$$This prior section gives the motivation, derivation, and examples of use of the compound interest formula.
In Precalculus, we now want to think about the result of adding in interest at shorter and shorter intervals: every hour, every minute, every second, every millisecond, and so on. That is, we will let $\,n\,$ (the number of times per year that interest is added in) go to infinity.
Letting $\,n\rightarrow\infty\,$ leads to socalled continuous compounding, which is the subject of the next section. It's another beautiful appearance of the irrational number $\,\text{e}\,$!