Hyperpolarization is a change in a cell's membrane potential that makes it more negative. It is the opposite of a depolarization.  It inhibits action potentials by increasing the stimulus required to move the membrane potential to the action potential threshold.
Hyperpolarization is often caused by efflux of K (a cation) through K channels, or influx of Cl (an anion) through Cl channels. On the other hand, influx of cations, e.g. Na through Na channels or Ca through Ca channels, inhibits hyperpolarization. If a cell has Na or Ca currents at rest, then inhibition of those currents will also result in a hyperpolarization. This voltage-gated ion channel response is how the hyperpolarization state is achieved. In neurons, the cell enters a state of hyperpolarization immediately following the generation of an action potential. While hyperpolarized, the neuron is in a refractory period that lasts roughly 2 milliseconds, during which the neuron is unable to generate subsequent action potentials. Sodium-potassium ATPases redistribute K and Na ions until the membrane potential is back to its resting potential of around –70 millivolts, at which point the neuron is once again ready to transmit another action potential.