This is not an explanation why the word tangent is used (I think the other answers nicely do this), but a historical digression.
The first occurence of a mathematical symbol as denotation of the tangent-function I have personally seen is in Leonhard Euler's "Introductio in Analysin Infinitorum, Volume 2" from 1748. This was written in Latin which at that time was the universal language of European scholars. The Latin word is "tangens" which is the present participle of the verb "tangere" and Euler abbreviated it by the symbol tang. In many modern languages the word "tangens" was replaced by "tangent" (English) or similar forms ("tangente" in French and Italian) , but for example in German and Russian the original form is used. Note that the "t" in the end of the word comes in via Latin declination.
Although the concept of tangent was known much longer (for certain in Arabic/Islamic mathematics, first formal use by Abū al-Wafāʾ who lived from 940 to 998), the word "tangens" was at first introduced by Thomas Fincke in his book "Geometria rotundi" from 1583.
From Euler's "Introductio in Analysin Infinitorum":
Edited: After a little "research" I found that the symbol tan for tangent was introduced long before Euler. In fact, it seems that Albert Girard introduced it in his work "Trigonométrie" from 1626. However, other sources say it was introduced by Edmund Gunter in 1624.
It may also be interesting that the Arabic word for tangent is "shadow" which is in use for more than 1000 years.