Q: What is wry tail?
A: A chicken whose tail always twists to one side has "wry tail." This is a condition that is exactly what it sounds like: your chicken's tail is held at a wry angle when in a relaxed position. It's a little crooked, like someone might have a crooked finger or a crooked nose. Wry tail is almost always genetic and not a cause for concern in terms of life quality. It causes no pain to your bird, and will not negatively affect your bird's life.
From above, you can see that this hen's tail twists to her right.
Even though wry tail will not negatively affect your bird, it is still considered a fault for exhibiting your birds at a poultry show. Additionally, because it is nearly always genetic you will probably not want to hatch eggs from a hen with this fault---or breed a rooster with this fault---as that could pass on the combination of recessive genes that makes wry tail express in your flock.
When is it not genetic? Again we can look at an analogy of a person with a crooked nose. You can be born with a crooked nose, or you might have been born with a straight nose that gets broken and heals crooked. So a bird can occasionally experience some injury or illness that would cause the appearance of a wry tail. That would be fairly unusual, and if you've had the bird from the time s/he was a chick, you would notice the change. If that change does happen suddenly, be sure to check your bird over for injury (or something like a cyst or tumor) near the tail. Unlike a genetic wry tail, of course an injury or illness might cause your bird pain, so be sure to check and make sure nothing is wrong.
If you receive a bird with a wry tail from a hatchery or breeder, most will appreciate receiving the report so they can try to remove the source of the recessive genes in the breeding flock.