Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Making contact at a single point or along a line; touching but not intersecting.
 adj. Irrelevant.
 n. A line, curve, or surface meeting another line, curve, or surface at a common point and sharing a common tangent line or tangent plane at that point.
 n. Mathematics The trigonometric function of an acute angle in a right triangle that is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
 n. A sudden digression or change of course: went off on a tangent during the courtroom argument.
 n. Music An upright pin in a keyboard instrument, especially in a clavichord, that rises to sound a string when a key is depressed and stops the string at a preset length to set the pitch.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A straight line touching a curve at a single point without crossing it there.
 n. In a right triangle, the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Symbols: tan, tg
 n. A topic nearly unrelated to the main topic, but having a point in common with it.
 n. A small metal blade by which a clavichord produces sound.
 adj. Touching a curve at a single point but not crossing it at that point.
 adj. Of a topic, only loosely related to a main topic.
from the GNU version of the Collaborative International Dictionary of English
 adj. Touching; touching at a single point.
 n. A tangent line curve, or surface; specifically, that portion of the straight line tangent to a curve that is between the point of tangency and a given line, the given line being, for example, the axis of abscissas, or a radius of a circle produced. See Trigonometrical function, under function.
from The Century Dictionary and Cyclopedia
 Touching; in geometry, touching at a single point: as, a tangent line; curves tangent to each other.
 n. In geometry: A straight line through two consecutive points (which see, under consecutive) of a curve or surface.
 n. The length cut off upon the straight line touching a curve between the line of abscissas and the point of tangency.
 n. In trigonometry, a function of an angle, being the ratio of the length of one leg of a right triangle to that of the other, the angle opposite the first leg being the angle of which the tangent is considered as the function.
 n. In the clavichord, one of the thick pins of brass inserted in the back ends of the digitals so that the fingers should press them against the strings, and produce tones.
 n. Any method of drawing a tangent to a curve.
 To bear or hold the relation of a tangent to.
 n. One of the keys or fingerlevers of the hurdygurdy.
 n. In railroading, a straight piece of track beginning and ending at a curve.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point
 n. ratio of the opposite to the adjacent side of a rightangled triangle
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

An argument can be made, but I think that the tangent is a bit too subtle for this discussion.

However, a tangent is also a powerful tool if used with skill.

The “Jebediah Whateley” tangent is completely gone.
Superhero Nation: how to write superhero novels and comic books » Version 1.1 is ready!

A tangent from a previous question: What's the one piece of music you never get tired of hearing?

The tangent is a side road that was not on your original plot map.

End of tangent from the theme of the thread, I hope.
The Volokh Conspiracy » Two Thirds of Americans Cannot Name Even One Supreme Court Justice

Following a tangent, is it just poker, or are IndianAmericans also underrepresented at the blackjack table?
Where Are All the Indian Poker Players?  Freakonomics Blog  NYTimes.com

On a similar tangent is Rocketboom (Joanne Colan's first vlog is up!) who is summed up brilliantly by Jeff Jarvis 'mate Fred Graver who is ex MTV, current VH1 dude who writes the best (non) post on the topic thru BuzzMachine 

And all this is a tangent from the main point: Nobody wants a democracy Iraqstyle with bombings, beheadings, and assorted sectarian violence.

Another interesting tangent is the way many leading or outspoken atheists are admitting that science/evolution/neodarwinism is not evidence for their atheism.
kirinqueen commented on the word tangent
in the communicative sense, of course.
October 17, 2008