Docking tails is a controversial matter today. Actual amputation of the tail is illegal in Britain, Norway, and Australia (it has been illegal in Britain for over 40 years, meaning that if you're writing something contemporary and you have a horse there with a docked tail...it had better be an import). It's now considered cruel.
This Clydesdale has had his tail docked extremely short. (Source: USDA).
The usual reason given for docking tails is to prevent the tails of working horses being caught in the machinery they are pulling. Another reason is to keep the lines of a driven horse from ending up under the tail. Tradition, however, is a big reason - in fact, in the 19th century, most riding horses also had their tails docked.
Most work horses these days don't have their tails amputated (although there are exceptions). Instead, the tail is kept trimmed very short to avoid the problems listed above. The actual bone of the tail is left intact, and the groom simply takes shears to it right below the tailbone (called the "dock" of the horse).
Many, especially in the UK, believe even that is inhumane (horses use their tails to keep the fly off). So, instead of docking, they put the tail in an updo. Polo players do the same thing, to keep the tail from hitting another pony or player in the face in tight quarters.
This polo pony's tail has been taped up. Draft horses often have elaborate braids for showing, but the end result is similar. Source: Wikimedia Commons.
Note that two harness breeds - the American Cream Draft and the Gypsy Vanner (more correctly called the Colored Cob or Irish Cob) are always shown with full, loose tails.
Finally, mules traditionally have their tails trimmed in layers. A straight trimmed tail indicates a green mule. One bang indicates a mule for pack usage, two bangs a mule suitable to be ridden by employees or guides, and three for mules suited for use by guests. This practice was created as a way of managing huge strings and is still perpetuated by the Grand Canyon mule stables.
Grand Canyon string mules - you can clearly see the three layers or "bangs" on the nearest mule's tail, indicating that this is a fully trained mule suitable to carry guests - even ones who can't ride.
Tomorrow I'll talk about why some horses have short manes.