EDIT Wrong name This character is actually not the real apostrophe. The glyph shown here and the glyphs on this codepoint in most fonts are actually the american minute-character. For example:

3'22" stands for 3 minutes, 22 seconds.

Incidentally, the character " (looking like double primes), that is representing seconds is frequently misused for quotation marks. The right ones here would be U+201D Right Double Quotation Mark and U+201C Left Double Quotation Mark
Often, this way of using those marks are relicts from the times when people needed workarounds for characters using typewriters or early computer programs.

The typographically correct shape for an apostrophe is not a straight upright line, but more of a comma at the height of the ascenders. It looks like U+2019 (Right Single Quotation Mark), or U+02BC (Modifier Letter Apostrophe)

 jsw  jsw 

EDIT Grammar In the English grammar, the apostrophe has three uses:
1. to form possessives of nouns (Tom\'s car)
2. to show the omission of letters (don\'t = do not, I\'m = I am)
3. to show some plurals of lowercase letters (like: p\'s and q\'s, but this rule really only applies to lowercase letters, no uppercase ones or any numbers).

It should not be used with posessive pronouns like his, hers, its, etc. because these already indicate a possession. Especially \"its\" and \"it\'s\" get confused regularly, but \"it\'s\" is an abbreviation of \"it is\" and doesn\'t mean \"...of it”

 jsw  : :  tuvalkin 

EDIT Number Separator In some locales, the apostrophe is used as a thousands separator. For example, the number ten thousand is written as 10\'000.

This is currently valid for i.a. parts of Switzerland, and reflects old-style or obsolete usage in many other countries, such as Portugal, where it was in use untill the 1940’ies

 jsw  tuvalkin 

EDIT ASCII In the ASCII character set, much of which was designed around typewriter keyboards, a number of characters were designed to represent multiple glyphs.

In English, this character is most often used as an apostrophe, thus the naming. It can also be used as both a beginning and ending single quote and the minute or foot symbol. This is why, in the majority of fonts, the tail points straight down, rather than curling in either direction.

 CurtisP