Compound interest simply means that interest
is earned on interest.
Compounding magnifies the impact that a given interest rate
has on the growth of principal.
Sometimes, as will be illustrated below, this impact
can be fairly dramatic.
There are two 'levers' by which compounding exerts its influence:

the compounding frequency or
interval (the number of compounding periods in a year); and
 the term (the number of years over which
the compounding takes place).
The more frequent the interval of compounding,
the greater the impact.
This is illustrated in the examples below.
However note that the compounding frequency "lever" is
subject to the law of diminishing returns.
Example.
What is the value of $100 invested for 1 year at 8%?
1. Annual Compounding. In this case there
is no compounding effect because the term is only
one year, the same as the compounding frequency.
Thus, all we have is simple
interest (i.e., the effective rate is equal to the
nominal rate)
2. Monthly Compounding. In this case there
are 12 compounding periods.
Interest earned each month is added to the
balance and is itself available to earn interest
in each succeeding month.
Thus, the future value is greater than the
amount calculated using annual
compounding.
3. Weekly Compounding. As should be expected,
increasing the frequency of the compounding period
increases the impact of the interest rate.
That it does so should be intuitive: more interest is
available sooner to earn more interest.
Whereas before we had to wait until the end of the month
before the interest was 'added back to the pot',
now it is being credited each week.
4. Daily Compounding. Now instead of earning
interest weekly, we earn it daily.
As expected the, the impact of the interest rate is
magnified.
However, this time the impact is not as dramatic
as might be expected.
5. Continuous Compounding. What if instead of days
we used an hourly compounding period?
Or minutes? Or seconds?
Better yet, what if the compounding period
were continuous?
It is possible to calculate such a compounding period
using the formula below.
(See Continuous Compounding
for more information.)
The result is the maximum effect that compounding frequency
can exert on a given interest rate and term.
Although the compounding is continuous, the result is
not much greater than what we got with
daily compounding. Which was not a whole lot
more than we got with weekly compounding.
This demonstrates, as mentioned earlier,
that while compounding frequency magnifies the
effect of a given interest rate, it is
subject to the law of diminishing returns.
Compounding exerts its most dramatic effect (for a given interest rate)
when the term is extended.
In other words, the longer an amount is subject to
compounding, the greater the effect.
Unlike compounding frequency, the impact of term length
is not subject to the law of diminishing returns.
Using the previous example of $100 invested at 8%, the
following calculations show the future value with
monthly compounding at 1, 5, 10, and 25 years.
After 1 year:
After 5 years:
After 10 years:
After 25 years:
The effect of term length is impressive: the
original sum doubles in less than 10 years and increases
more than seven fold in 25 years.
These results are even more impressive
when you consider that this example
illustrates the growth of a single sum;
there is no additional principal being added,
only the crediting of periodic interest earned.
Thus, when we speak of the miracle of compounding
for a given interest rate, we are talking more in terms of
the impact of the term (the length of time
an amount is subject to compounding) than the
frequency (the number of
compounding periods per year).
In fact, if we had used continuous in the last
example where the term was 25,
the future value would have been only 738.91,
an increase of less than 1% over the 734.02
earned with monthly compounding.