In order to understand the physiological result of chloride channels opening in neurons, one must first understand how action potential generation occurs. Action potentials are short electrical events that generate signals, which can then be communicated to other neurons at meeting points called synapses. These signals are passed on from one neuron to the next via the release of chemicals known as neurotransmitters from the axon of the signal generating neuron. These neurotransmitters act on specific receptors located on dendrites of the receiving neuron. Many neurotransmitters exist, and each type can activate a specific channel or set of channels leading to either the generation of another action potential or the inhibition of further signaling.
Action potentials are generated via changes in the electrochemical gradient of neurons. This gradient is generated by differences between the intracellular and extracellular concentrations of key ions such as sodium, potassium, calcium, and chloride. In neurons this resting potential typically lies around -70 millivolts (mV). The concentration for sodium is higher outside of the neuron (about 150 milliMolar (mM)) and lower on the inside of the neuron (about 15mM). The opposite is true in the case of potassium with the internal concentration being higher (about 150 mM), while the external concentration is lower (about 5.5 mM). Since neuronal membranes are considered leaky, ions are able to diffuse along concentration gradients towards the side of the neuron that has a lower concentration. Because of this, neurons rely on the sodium/potassium ATPase exchange pump to maintain resting potential. This is done via the exchange of 3 sodium ions being pumped out the cell for 2 potassium ions being pumped into the cell and requires ATP for activation.
In order for an action potential to be generated the membrane potential must typically be raised to -55 mV. This is done through the activation and opening of channels, via neurotransmitters, that conduct positive sodium ions into the cell. It is at this point, referred to as the "threshold potential," that depolarization occurs and an action potential is generated. This point is also typically referred to as the point of no return. Upon reaching a membrane potential of -55mV, voltage gated channels located on the cell surface of the neuron that are responsible for allowing sodium into the neuron begin to open. This opening leads to a large influx of positively charged sodium ions into the neuron causing an action potential to be generated. After a quick period, slow opening potassium channels begin to allow potassium ions to flow out of the cell, thus decreasing the amount of positive charge inside the neuron and restoring the resting membrane potential, allowing later action potentials to be generated.
In the specific case of chloride ions, these ions are present in a much lower concentration inside the neuron (around 4 mM) than outside the neuron (around 100 mM). Unlike sodium and potassium, chloride is a negatively charged ion. Therefore, in the resting state chloride defuses down its concentration gradient and into the cell. In some cases, chloride ions are pumped back across the cell surface to maintain resting membrane potential. In the specific case of a chloride channel opening (presumably through neurotransmitter activation), the influx of chloride into the neuron would decrease the resting membrane potential driving it more negative than -65mV. This is referred to as hyperpolarization. This phenomenon is known as an inhibitory postsynaptic potential. As the name implies, and because threshold potential does not change, this chloride ion influx makes it more difficult for depolarization of neurons, and for action potential firing to occur. Hope this helps!
Typically when discussing the action potentials that occur in neurons, either to send messages to other neurons, trigger release of products from glands, or to trigger muscle contraction, one considers two ions: sodium (Na+) and potassium (K+). Na+/K+ pumps use ATP to set up a concentration gradient on either side of the cell membrane with sodium as the dominant extracellular ion and potassium dominant inside of the cell. Negatively charged molecules inside the cell create a negative charge on the inside of the cell relative to the outside. That is how the potential of the membrane is measured; it is always the inside relative to the outside. The resting membrane potential (before any messages are sent) is again due to the Na+/K+ gradient and the negatively charged molecules inside and can typically be measured at -70 mV.
When an excitatory message(s) of sufficient strength is sent, the Na+ channels open and Na+ diffuses down its concentration gradient entering the cell. This makes the membrane positive now (since Na+ have a positive charge and charge is always inside relative to outside). This change opens K+ channels allowing K+ to exit the cell. Losing these positive charges brings the membrane potential back to negative. In fact, the channels stay open a bit too long and the potential goes more negative than resting membrane potential. This increased negativity makes it much more difficult for incoming signals to meet the threshold needed to trigger the opening of the Na+ channels described above. This is called hyperpolarization and prevents another action potential from occurring immediately.
The opening of chloride (Cl-) channels creates a similar situation to the K+ channels being open too long. Cl-, like Na+, is a dominant ion outside of the cell. When Cl- channels open, negative ions move into the cell making the membrane potential even more negative than resting membrane potential. Again, this is called hyperpolarization. This makes it very difficult for other neurons to trigger an action potential in the given neuron. Thus, anything that would cause the opening of Cl- channels in a neuron would be inhibitory and would potentially prevent the receiving neuron from firing action potentials. GABA is an inhibitory neurotransmitter that results in the opening of Cl- channels and an influx of Cl- ions into the neuron.
This higher-level question implies that you already know the basics of how nerve cells operate--that nerve cells carry messages to other parts of the body both electrically, along the axon, and chemically, across the gap between nerve cells. This gap is called the synapse. Nerve cell "firing" happens when the difference in electrical charge between the inside and the outside of the neuron's cell membrane is great enough. The nerve cell can then "fire"; this is an all-or-nothing response.
Chloride channels, when opened, appear to be one of the ways that positively charged ions are moved inside the cell, and negatively charged ions out, until the charge difference is great enough that the nerve impulse can be generated. This is an area currently under study, as problems with the channels may be implicated in seizure diseases such as epilepsy. The links below give a lot more detail.