Comets
At the heart of a comet is its small nucleus (0.5 - 50 Km across - in comparison to the Earth's 12,756 Km), composed of dust embedded in frozen volatile materials (water, methane, ammonia, carbon dioxide, carbon monoxide and other gases) that orbits the Sun typically in a highly elliptical orbit (squashed, rather than circular - with the closest and furthest distances from the Sun being very different).
As it approaches the Sun some of these volatile materials evaporate off the nucleus of the comet to form its coma - a very large (100,000s Km across) diffuse, dusty "atmosphere" that is steadily streams off into space (the comet's gravity being insufficient to hold on to it). Very near the Sun the coma is often blown away by solar radiation to form the dust tail - with the solar wind dragging ionized gas away in a slightly different direction to form the plasma tail. Without this (visibly lost) volatile material it wouldn't be called a comet - but an asteroid. Some asteriods are old comets that have lost all their volatile materials through too many trips passed the Sun.
When it's more distant from the Sun (as it is for the vast proportion of the time) the comet dims dramatically as the evaporation to form the coma slows and it is less illuminated. At sufficient distances from the Sun the evaporation ceases and a passing observer would see just its sooty dark nucleus.
Comet orbital periods vary from just a few years to over 100,000 years. Most comets are discovered as they approach the Sun and become visible (to telescopes). Small comets are quite common, with about a dozen discovered every year, but most of these aren't visible to the naked eye. Some of us can look forward to the return of medium/short period dramatic comets like Halley (period: 76 years - next due in 2061) and about once a decade a spectacular longer period comet like Hyakutake (period 29,500 years) or Hale-Bopp (period 2 380 years) turns up unannounced from the outer solar system.

Links:
More about comets:
zoomschool comets
"nine planets" comets

Astronomy Notes on comets...
comets
comet orbits - Oort Cloud and Kuiper Belt
comet beginnings and ends
review questions

ScienceWeb...
teachers tips
comet questions

Windows to the Universe comet table
Why do published comet orbits change 

Halley's comet...
The return of Halley's comet
"nine planets" Halley' comet

comet Hale-Bopp...
Teacher's newsletter preview
Changing estimates of orbit data
3D view of orbit

current comets...
Cometography.com current comets

Sizes of objects in the Solar system...
"nine planets" Solar system data