Brian,
The reason that you burned out the Arduino pin is that you didn't have a current limiting resistor on the Arduino side. Take circuit B. When Arduino pin 3 goes high, current is free to not only flow from the emitter (arrow part) to the collector (the current you want, going to the load) but also from the emitter to the base (which connects to the Arduino). Without a current limiting resistor, you are trying to have the Arduino sink an amount of (negative) current limited by the resistance of the wires and the tiny semiconductor junction. This will fry the Arduino pin very quickly.
Going back to circuit B, you need to put about a 1k resistor between the Arduino and the transistor. That will limit the current and save the Arduino from damage.
The transistor allows you to control a large current (emitter to collector) with a small current (emitter to base), but you have to make sure that the emitter to base current is small. Putting a current limiting resistor in there will do just that.
The fundamental problem is that if you connect any constant voltage source (battery, power supply, Arduino output) to ground with a nearly zero resistance connection, nearly infinite current will try to flow. Nearly infinite currents tend to make things go snap crackle pop and then stop working.
Hope this helps.
Regards,
Martin