Question: I have permission to access a (borrowed) Arduino Uno, and the documentation says it can be powered with 3.3 Volts from the 3V3 pin:

3V3. A 3.3 volt supply generated by the on-board regulator. Maximum current draw is 50 mA. 

<http://arduino.cc/en/Main/arduinoBoardUno>

However, I would like to use a 24V DC power source. 

The safe range is supposed to be 6V-20V.

If I could get a transformer in place to change the 24V DC to 8V DC, could I apply that 8V to the 3V3 input (without frying the board)?

Answer: 
Connecting anything much above 3.3V (as in say 3.6V) to that rail would fry the Arduino. A transformer only operates with AC voltage not DC, so the best way for what you want to achieve is a step-down regulator also known as a buck converter. Variable ones that can accept 24V DC and output 3.3V (be careful to adjust it before connecting) are available on e-bay. Here's the first example I found:

[DC Adjustable Voltage Buck Converter 4.5-25.5V to 3-24V Step-down Power Module](http://www.ebay.com.au/itm/DC-Adjustable-Voltage-Buck-Converter-4-5-25-5V-to-3-24V-Step-down-Power-Module-/400432774838?pt=AU_B_I_Electrical_Test_Equipment&hash=item5d3ba73eb6)

**But important:** I'd taken at face value that your statement was correct that it was OK to power the board from 3.3V over that line. As [Peter Bennet](https://electronics.stackexchange.com/users/19835/peter-bennett) correctly states some parts are actually 5V based and the 3.3V rail is the output of a regulator. Without removing the regulator and knowing the exact consequences that isn't a good idea.

Instead you should consider setting the regulator for say 8V and connect it to the 6V-20V input socket on the Arduino where you'd normally apply an external power supply / wall wart etc. That gives you a bit of extra safety if the adjustment on the regulator ever gets changed or drifts a bit.
The board has a power supply input named X1 , the voltage that is connected there goes through a [NCP1117](http://www.onsemi.com/pub/Collateral/NCP1117-D.PDF) regulator to generate the 5v supply that feeds the 5v parts

![enter image description here](https://i.stack.imgur.com/bNsoC.jpg)

and the 5v supply is then fed to another regulator [LP2985-33](http://www.ti.com/litv/slvs522n) that outputs the 3.3v in order to supply the 3.3v parts

![enter image description here](https://i.stack.imgur.com/wWIH0.jpg)

Apart from that, there is an available input that is intended to supply the board from the a USB cable (5v)

![enter image description here](https://i.stack.imgur.com/8P5tM.jpg)

According to the board specs for the supply input X1

> 
> Input Voltage (recommended) 7-12V  
> 
>  Input Voltage (limits) 6-20V
> 
> 
> 

The high input voltage limit of 20v is forced by the maximal input voltage of the NCP1117, a higher voltage would damage the regulator.

Basically you can only connect voltages to two points, one is the USBVCC which i intended for regulated 5v supply (from USB port or 5v wall supply) and the other one named X1 where you should connect a supply that is between 6v-20v and which is where you should connect the 8v or 12v (referring always to DC voltages) you talk about.  

There is no 3.3v input.