user72442: Is there a way for me to have this work the opposite way?
I have one cable with an alligator clip hooked up to the reset pin and another hooked up to ground.
When I clip them both together, the board resets.
However, I would like for this to work in the opposite way such that they’re already clipped together and when you detach the cables, the board resets.

user42015: The easiest way to do that would be to use an external pulldown resistor (a resistor from the reset pin to ground) and connect to Vcc (5V on the Nano) via your alligator cables.
As long as the alligator cables are connected, the reset pin is held HIGH through them. When you disconnect them, the pulldown resistor pulls the reset pin to LOW. That holds the Nano in reset until you again connect the alligator clips.
If you want the Nano to reset and then directly work again without the alligator clips reconnected, it gets more difficult. One way I can think of is building a monostable vibrator circuit, which gets in the excited state, when your alligator clips are disconnected, resets the Arduino and then goes back into the stable stable, letting the Arduino execute its code. Though that is more an electrical question, than an Arduino one.

user4143: My personal preference would be to attach a NOT gate to the RESET pin.  This can be constructed very simply from a single NPN transistor (or N-channel MOSFET) and a resistor.

simulate this circuit – Schematic created using CircuitLab
With the wire disconnected R1 turns on Q1 which pulls the RESET pin LOW.  With the wire connected Q1 is turned off, allowing RESET to be pulled back up by the pullup resistor that's built into the Arduino.
This method also won't interfere with the programming reset action of the Arduino.

72442: <p>Is there a way for me to have this work the opposite way?</p>
<p>I have one cable with an alligator clip hooked up to the reset pin and another hooked up to ground.</p>
<p>When I clip them both together, the board resets.</p>
<p>However, I would like for this to work in the opposite way such that they’re already clipped together and when you detach the cables, the board resets.</p>
 42015: <p>The easiest way to do that would be to use an external pulldown resistor (a resistor from the reset pin to ground) and connect to Vcc (5V on the Nano) via your alligator cables.</p>
<p>As long as the alligator cables are connected, the reset pin is held HIGH through them. When you disconnect them, the pulldown resistor pulls the reset pin to LOW. That holds the Nano in reset until you again connect the alligator clips.</p>
<p>If you want the Nano to reset and then directly work again without the alligator clips reconnected, it gets more difficult. One way I can think of is building a monostable vibrator circuit, which gets in the excited state, when your alligator clips are disconnected, resets the Arduino and then goes back into the stable stable, letting the Arduino execute its code. Though that is more an electrical question, than an Arduino one.</p>
 4143: <p>My personal preference would be to attach a NOT gate to the RESET pin.  This can be constructed very simply from a single NPN transistor (or N-channel MOSFET) and a resistor.</p>

<p><img src="https://i.stack.imgur.com/XgkOA.png" alt="schematic" /></p>

<p><sup><a href="/plugins/schematics?image=http%3a%2f%2fi.stack.imgur.com%2fXgkOA.png">simulate this circuit</a> &ndash; Schematic created using <a href="https://www.circuitlab.com/" rel="nofollow">CircuitLab</a></sup></p>
<p>With the wire disconnected R1 turns on Q1 which pulls the RESET pin LOW.  With the wire connected Q1 is turned off, allowing RESET to be pulled back up by the pullup resistor that's built into the Arduino.</p>
<p>This method also won't interfere with the programming reset action of the Arduino.</p>
