user1333: (I have a Sparkfun RedBoard, but this question seems to apply to R3 Unos and Uno-compatible boards.) As I was building the first circuit in my SIK guidebook (add a resistor, LED, hook it up and make it blink from code), I noticed a blue LED on the board itself did everything that the circuit's LED did - blinking according to the programming. 
Why is this LED here? What use cases is it for?
What kind of circuits will I have to adjust to account for this LED?
Is pin 13 traditionally a debug or a troubleshooting pin? Where did this convention come from?

user881: Yes, pin 13 is traditionally an output pin that drives an LED. But the blue LED should not be blinking as it is the "power on" indicator - do you have a grunty power supply to your board?
You can experiment with PWM to the pin13 LED, which will control its apparent brightness. There are plenty of example sketches on the web.

user973: What everyone else said, plus: if you find the digital-13 LED gives you a problem, you can always remove it from the circuit, either by removing the LED, or the series resistor. 
This might be useful if you are running out of digital I/Os, or you want several contiguous I/O bits and don't want one of them to behave differently from the others. 

user11: 
What kind of circuits will I have to adjust to account for this LED?

You'll need to take it into account if you're using that pin as a digital input.

NOTE: Digital pin 13 is harder to use as a digital input than the
  other digital pins because it has an LED and resistor attached to it
  that's soldered to the board on most boards. If you enable its
  internal 20k pull-up resistor, it will hang at around 1.7V instead of
  the expected 5V because the onboard LED and series resistor pull the
  voltage level down, meaning it always returns LOW. If you must use pin
  13 as a digital input, set its pinMode() to INPUT and use an external
  pull down resistor. Arduino - DigitalPins


user84: The LED on pin 13 is used by the optiboot loader (the one used on UNO):

at Arduino boot time (the LED blinks a few times)
when uploading a sketch to Arduino

I haven't checked other bootloaders, they may provide the same behavior as the optiboot.
For optiboot, there are optional defines (at compile time) to modify this behavior:

LED_START_FLASHES defines the number of flashes of pin 13 LED at boot time (can be set to 0)
LED_DATA_FLASH will use pin 13 LED during sketch upload if defined at compile-time

These defines are explained in hardware/arduino/bootloaders/optiboot/optiboot.c from within your Arduino IDE install directory.
If you want to change these, you will have to recompile the optiboot loader first and then burn it to your Arduino through an ISP programmer.
I guess one other reason for the LED on pin 13 was to simplify the demonstration of the "Hello World" sketch for Arduino, namely the Blink sketch, without the need for any extra component.

1333: <p>(I have a Sparkfun RedBoard, but this question seems to apply to R3 Unos and Uno-compatible boards.) As I was building the first circuit in my SIK guidebook (add a resistor, LED, hook it up and make it blink from code), I noticed a blue LED on the board itself did everything that the circuit's LED did - blinking according to the programming. </p>

<p>Why is this LED here? What use cases is it for?</p>

<p>What kind of circuits will I have to adjust to account for this LED?</p>

<p>Is pin 13 traditionally a debug or a troubleshooting pin? Where did this convention come from?</p>
 881: <p>Yes, pin 13 is traditionally an output pin that drives an LED. But the blue LED should not be blinking as it is the "power on" indicator - do you have a grunty power supply to your board?</p>

<p>You can experiment with PWM to the pin13 LED, which will control its apparent brightness. There are plenty of example sketches on the web.</p>
 973: <p>What everyone else said, plus: if you find the digital-13 LED gives you a problem, you can always remove it from the circuit, either by removing the LED, or the series resistor. </p>

<p>This might be useful if you are running out of digital I/Os, or you want several contiguous I/O bits and don't want one of them to behave differently from the others. </p>
 11: <blockquote>
  <p>What kind of circuits will I have to adjust to account for this LED?</p>
</blockquote>

<p>You'll need to take it into account if you're using that pin as a digital input.</p>

<blockquote>
  <p>NOTE: Digital pin 13 is harder to use as a digital input than the
  other digital pins because it has an LED and resistor attached to it
  that's soldered to the board on most boards. If you enable its
  internal 20k pull-up resistor, it will hang at around 1.7V instead of
  the expected 5V because the onboard LED and series resistor pull the
  voltage level down, meaning it always returns LOW. If you must use pin
  13 as a digital input, set its pinMode() to INPUT and use an external
  pull down resistor. <sub><a href="http://arduino.cc/en/Tutorial/DigitalPins">Arduino - DigitalPins</a></sub></p>
</blockquote>
 84: <p>The LED on pin 13 is used by the <strong>optiboot</strong> loader (the one used on <strong>UNO</strong>):</p>

<ul>
<li>at Arduino <strong>boot time</strong> (the LED blinks a few times)</li>
<li>when <strong>uploading a sketch</strong> to Arduino</li>
</ul>

<p>I haven't checked other bootloaders, they may provide the same behavior as the optiboot.</p>

<p>For optiboot, there are optional defines (at compile time) to modify this behavior:</p>

<ul>
<li><code>LED_START_FLASHES</code> defines the number of flashes of pin 13 LED at boot time (can be set to <code>0</code>)</li>
<li><code>LED_DATA_FLASH</code> will use pin 13 LED during sketch upload if <strong>defined</strong> at compile-time</li>
</ul>

<p>These defines are explained in <code>hardware/arduino/bootloaders/optiboot/optiboot.c</code> from within your Arduino IDE install directory.</p>

<p>If you want to change these, you will have to recompile the optiboot loader first and then burn it to your Arduino through an <strong>ISP programmer</strong>.</p>

<p>I guess one other reason for the LED on pin 13 was to simplify the demonstration of the "<em>Hello World</em>" sketch for Arduino, namely the <code>Blink</code> sketch, without the need for any extra component.</p>
