How can I null the output of this circuit when a switch is activated?

Specifically, how would I cancel the Redstone signal going to the stone block? What type of logic gate would I need?

I want to be able to active a switch on the right input (in the screenshot) that will null the resulting output:

Based on your description, you are looking for either a NAND gate, a XOR gate, or an IMPLIES gate. Which one you choose will depend on how you want the logic gate to behave when the main (A) circuit is off.

NAND gate:

• A is on and B is on = off
• A is on and B is off = on
• A is off and B is on = on
• A is off and B is off = on

XOR gate:

• A is on and B is on = off
• A is on and B is off = on
• A is off and B is on = on
• A is off and B is off = off

A IMPLIES B (inverted output)

One last possibility for completeness sake is the A implies B circuit, although for the purposes of your desired output I'm inverting the output.

• A is on and B is on = off
• A is on and B is off = on
• A is off and B is on = off
• A is off and B is off = off

While I'm not too happy with the XOR gate demonstrated (it seems to be comparatively large and clunky), the Minecraft Wiki article does demonstrate a few variants for each logic gate.