Why do nerve impulses only travel in one direction?

GFP expressing pyramydal cell in mouse cortex. Wei-Chung Allen Lee, et al.

Neurones (or nerve cells) are said to "fire" when an electrical impulse travels along the length of a neurone. But what triggers it and how does it spread?
The fluid inside and outside a cell contains many different kinds of charged particles called ions. Some are negatively charged and some positively. Two important ones are Na+ (sodium) and K+ (potassium).
The resting state
When the neurone is in a resting state it is actually working quite hard. The cell membrane has a structure call a sodium-potassium pump which pumps Na+ to the outside and K+ to the inside of neurones. Because Na+ cannot readily diffuse across the cell membrane a higher concentration of Na+ builds up on the outside of the cell. This redistribution of ion together with the ones from other ions and proteins results in a charge being formed across the membrane - the inside is more negatively charged.

Firing off
When a neurone is stimulated - from an adjacent neurone or an external stimulus (heat, touch) - changes take place in the membrane. Sodium channels in the membrane open briefly and Na+ floods into the cell interior. The inside of the cell changes from negatively charged to positive. This process is known as depolarisation.

Recovering
Just as the sodium channels snap shut potassium channels open and K+ diffuses out of the cell. This causes the charge the inside the cell to head back towards negative again. The sodium-potassium pump works hard and restores the Na+ and K+ to their original sides of the membrane in resting state.  This process is known as repolarisation.
This electrical event is called an action potential.

The refractory period
So the cell membrane goes through a cycle of depolarisation and repolarisation. If another stimulus arrives at the membrane during this cycle, the area cannot depolarise again. The current cycle has to finish before it is capable of depolarising again. The refractory period refers to this time when the membrane is not susceptible to depolarising.
And here's why it travels in one direction
An action potential starts at on end of a neurone and spreads as a wave along the it. The currents flowing inwards at one point on the neurone also depolarises the adjacent sections of the membrane. Once an action potential has occurred at a patch of membrane, the membrane patch needs time to recover before it can fire again (the refectory period). For this reason, only the unfired part of the membrane can respond with an action potential. The part that has just fired is unresponsive until the action potential is safely out of range.
Typically the action potential starts at the axon end (by stimulation from another nerve) and travel along a neurone to the synapse end.
Clinical implications
Local anaesthetics are membrane stabilising drugs;. They decrease the rate of depolarisation and depolarisation of neurones. These drugs act mainly by inhibiting sodium influx in the neuronal cell membrane. When the influx of sodium is interrupted, an action potential cannot arise and signal conduction is inhibited.
Disturbances of sodium or potassium levels in the body can directly affect these chemical processes and will interfere with normal nerve conduction.
Low serum potassium levels will often cause a generalised weakness. Increased potassium levels interferes with depolarisation of cells resulting in inability of nerves to fire.
High levels of sodium will cause weakness initially and then with more severe elevations of the sodium level, seizures and coma may occur.