An action potential (often called a "spike") is a brief change in voltage that travels along the cell membrane of a neuron. An action potential is typified by the depolarization of a membrane which has a negative resting potential.
One of the most important ways by which neurons in the brain carry information is by means of electrical signals. At rest, neurons have a 'resting membrane potential' of typically about -70 mV: this is the potential difference between the inside of the cell and the external environment. However, this membrane potential is continually being disturbed by the chemical signals (neurotransmitters) released by other neurons. In the brain a neuron might be receiving inputs from as many as ten thousand other neurons, these inputs usually consist of brief excitations (EPSPs; excitatory post-synaptic potentials) or of brief inhibitions (IPSPs; inhibitory post-synaptic potentials) that either make the membrane less negative (depolarisations) or more negative (hyperpolarisations). If a neuron becomes sufficiently depolarised, perhaps because of a flurry of EPSPs, then the membrane potential may reach the critical threshold for triggering an action potential (commonly called a spike). An action potential is a very large and rapid rise in the cell membrane potential, that lasts for only about one millisecond before the membrane potential returns again to around its resting potential.
Action potentials generated in neurons travel along axons to their nerve endings, depolarisation of which results in the release of neurotransmitters, which cross the cleft (synapses) between the nerve endings and other neurons. Action potentials are thus a major element involved in the transmission of information in the brain and peripheral nervous system.