EDIT History
The first known examples of asterisks are very old, from the Sumerians of about 5000 years ago.

The asterisk we know today derived from the times when having a family trees printed was a necessity for noble families and the printers needed a symbol for birth. The asterisk used to have six arms each of which had a drop-like ending. In this fashion, the asterisk appeared on typewriters as well, but often it would not print cleanly.

Apparently, many Arab people would also not buy Typewriters with this symbol on it since its appearance reminded them of the Star of David on the Israeli flag.

Hence the Asterisk was put onto many systems as a five-stared symbol, and without drop-endings. There is a five-star of very similar appearance encoded in the Arabic Block as U+066D Arabic Five Pointed Star.
 
EDIT Footnotes The asterisk is used for footnotes, just like the dagger often is in anglo-american use. The use of more than three asterisks should be avoided.
If the reference is made in relation to a single* word, the asterisk is positioned directly behind it. For a reference in relation to part of a sentence, it is positioned behind the comma,** and for a whole sentence after the finishing punctuation mark.***

*like this one
**or this bit
***see?
 
EDIT Typography
The asterisk is put directly behind the word without a space, but sometimes manual adjustment is necessary if it touches the final letter, like here f*
When used for a birthdate, the asterisk is put directly in front of the date: *07.01.1966
but when a word is in front of the date, a space is used: * around 1960
 
EDIT Linguistics
An asterisk can either indicate that a word is a form that was historically reconstructed (in historical linguistics) or that a word or phrase is ungrammatical (in theoretical linguistics)
 
EDIT Mathematics
Sometimes used as symbol for multiplication in older manuscripts (first with Johann Rahn in »Teutsche Algebra«, 1659). You can use Google as acalculator and use it a symbol for multiplication.
It signifies the complex conjugate of a complex number, e.g. if x=a+ib, x*=a-ib. It is also used as an infix binary operator (an operator that is positioned between the two numbers it operates with, a style familiar to all of us)
The lowered asterisk is a symbol for convolution, e.g. f ∗ g is a convolution of f with g.
 
EDIT Informatics
In some computer interfaces like the Unix Shell or Microsoft's Command Prompt, the asterisk is the wildcard character that represents any string of characters.
It also has many different uses in various computer languages like C or Perl
 
EDIT Internet
Often used in online communucation like emails and IRChats to indicate emotional statements »*grin*«, »*smile*« or for emphazising certain statements (because writing all-caps is usually considered rude): »In a character encoding, the character is *not* the same thing as a text string of length 1«.
 
EDIT Other uses
Genealogy: Date of birth
Chemistry/Physiks: radioactivity
Meteorology: snow
Music: »pedal up«, a sign in sheet music to let go of the pedal. It has a seperate codepoint in Unicode 4.1 under U+1D1AF
Botany: has been used for subspecies
Mushrooms: in books about mushroom picking an asterisk usually means edible. Two asterisks mean good, three, very tasty

 
EDIT Wild card for Google
When a search on Google is made with a whole sentence, an asterisk can be inserted to replace a word within the sentence to allow for a wider variety of results, e.g. searching for "spaghetti is the most * dish on earth" will return results with "popular dish", "boring dish" etc.
 
EDIT Security measure
Often when a password is entered in a terminal or online, the characters are not displayed and replaced with an asterisk.

 
EDIT Special key on telephone keypads, to the left of the zero. The other one is the hash-key.
EDIT It should also be mentioned the asterisk\'s use of formally censoring certain, mostly four letter words: f***
 
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U+002A ASTERISK
DEUTSCH : ENGLISH