Question

How can an action potential in one neuron cause a second neuron to be inhibited?

Answer 1

When the Neurotransmitter excreted triggers an IPSP.

Explanation:

Neurons are arranged in Neural Networks.
These can have various forms, although a few "classes", or "standard" structures have been identified.

In most of these, one neuron excites, or tries to excite, the next one (chemical triggering is a one-way process). This is known as an EPSP, or Excitatory Post Synaptic Potential.

Sometimes though, the signal transmittted needs to have the opposite effect,and triggers an IPSP: Inhibitory Post Synaptic Potential.

Some examples of Neural Networks (These are called microcircuits):

In these examples, the Red neurons trigger an IPSP in the receiving neuron: In example B for instance, the left green neuron triggers an EPSP in the red neuron in the middle, which in turn gives an Inhibitory siglnal to the third neuron.
In the Recurrent Inhibition Feedback example ( E1), The second neuron, which apparently doesn't like to be triggered, loops back to its Excitator via an "inhibitory" neuron, telling it to stop.....

Whether the signal carried across the (chemical) synapse is Excitatory or Inhibitory depends on the Neurotransmitter used:
There are around 90 identified, and most have an Excitatory effect.
Some though are inhibitory. Foremost amongst these is GABA (Gamma-Amino Butyric Acid). Serotonin and, apparently,Melatonin are two others....

pics, as usual, courtesy of University of Texas, mcGovern Medical School: http://neuroscience.uth.tmc.edu/s1/introduction.html