This article is from the Bicycles FAQ, by Mike Iglesias with numerous contributions by others.
Chain life is almost entirely a cleanliness and lubrication question
rather than a load problem. For bicycles the effect of load
variations is insignificant compared to the lubricant and grit
effects. For example, motorcycle primary chains, operated under oil
in clean conditions, last as much as 100,000 miles while exposed rear
chains must be replaced often.
The best way to determine whether a chain is worn is by measuring its
length. A new chain has a half inch pitch with a pin at exactly every
half inch. As the pins and sleeves wear, this spacing increases,
concentrating more load on the last tooth of engagement, changing the
tooth profile. When chain pitch grows over one half percent, it is
time for a new chain. At one percent, sprocket wear progresses
rapidly because this length change occurs only between pin and sleeve
so that it is concentrated on every second pitch; the pitch of the
inner link containing the rollers remaining constant. By holding a
ruler along the chain on the bicycle, align an inch mark with a pin
and see how far off the mark the pin is at twelve inches. An eighth
of an inch (0.125) is one percent, twice the sixteenth limit that is a
prudent time for a new chain.