Electronic – arduino – How to use external power supply for Arduino Uno

You have a couple of easy options here: feed in an unregulated voltage, like the 12v, to the Arduino's Vin pin (on the Arduino headers), and let the Arduino regulate it, or feed in 5v to the 5v pin. In either case, of course, you should connect the ground pin to your PSU's ground rail. From the Arduino schematic, you can see that Vin is only used to feed the 5v regulator and the comparator for the USB/Vin switch, so if you power the Arduino over the 5v rail, you should not connect anything to the USB port.

Assuming you have a fairly stable and clean 5v source, feeding the 5v is the more efficient option.

You are correct that the Arduino is powered over USB only when no DC supply is present. This is implemented with a comparator, comparing half the voltage on the Vin pin to the 3.3v rail - so the switch will work correctly as long as power is supplied to Vin.

I would guess you might have had a failure on one of power supply components. Here's the schematic for your board: http://arduino.cc/en/uploads/Main/Arduino_Uno_Rev3-schematic.pdf

Here are a few thoughts:

1. What is the actual voltage on your 5V battery? If it's not fully charged, it may not have enough voltage to keep everything running.
2. Check the voltage across fuse F1 as you plug in the battery... Perhaps the battery is allowing for a higher inrush current than the USB bus, causing this fuse to open? This doesn't seem likely as there's not that much capacitance on the board, but is easy to check.
3. It's possible it's not a power supply problem at all... can you clarify what you mean by "it doesn't work"? Does the board appear to power? It could be that you're not properly grounding your serial connection to whatever you're communicating with, and having it plugged into your computer via USB is providing an alternate path for the serial ground and causing serial communications to work. Without more knowledge of what you're communicating with and how it's connected, however, this is nothing more than a wild-ass guess.