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[on:]

The Atmega328 chip can use 5V, thus it makes sense to use the USB-charging-method, because USB protocol provides 5V power.

So why is it that the Arduino when powered externally, say, by a battery needs a 9V power supply?

Given that one can additionally use a transformer/regulator/etc. combination to in turn change the voltage the device finally gets, HOW does one go about actually choosing what voltage to power your project with?

For example, what voltage battery would you choose in the following two contrasting cases?
Atmega-chip+one-LED (simple) ---versus--- Atmega-chip+Xbee+LCD+buzzer (more power-demanding)

[Reply #1 on:]

The ardu board has one on board regulator 7805, this regulator  need in the input more than 7,5 volts,so in the battery connetor
you can connect from 7,5 volts till 20, so can use 9 volts battery or 12 volts battery

[Reply #2 on:]

The ardu board has one on board regulator 7805, this regulator

Well, it is a 5V regulator, but it is not a 7805.  The modern Arduino boards use a LDO regulator.  While 7-12V is recommended, you might find in low-current applications you can go down to 6-6.5V

what voltage battery would you choose in the following two contrasting cases?
Atmega-chip+one-LED (simple) ---versus--- Atmega-chip+Xbee+LCD+buzzer (more power-demanding)

There is a lot of "it depends" in this question.  Most people would not choose a standard 9V battery, its current capacity is far to low (as compared to AAs.)  Voltage is rarely as important as current capacity. 

How long do you want your project to run?  What is it doing when it is running?  How much space do you have available?  What is your budget?  Can the batteries be re-charged?

Questions such as that will answer the question.

[Reply #3 on:]

The answer in part is so that you have more choice how to power it. Rather than have to give it exactly 5V, you can give it anything between 7.5 and 20 V.