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The symbol (|) has various names that refer to differing, yet sometimes related semantics: One of the more popular names is the Sheffer stroke, though often referred to as a pipe (by the Unix community) and Vertical bar, verti-bar, vertical line or divider line by others.
Broken bar (¦) is a separate character.
The typical keyboard layout used in the United Kingdom features separate keys for vertical bar and broken bar, however most UK keyboard drivers today map both keys onto the vertical bar, as broken bar has hardly any practical application. Keyboards in other countries do not make this distinction, and there, the vertical bar character is often depicted as a broken bar, to distinguish it from the "I" (uppercase "i") character.
Etymology
The name pipe for the character "|" was a natural consequence of its usage in Unix shells, where it is used to notate the usage of the Unix pipeline construct.
Usage
Mathematics
The vertical bar is used as a mathematical symbol in
absolute value: | x | , read "the absolute value of x".;
norms:  norm of x sub one, x sub two"; though Unicode also provides a special double vertical line symbol U+2016: ‖x‖;
set definition: {x | x < 2}, read "the set of x such that x is less than two". Often sets are written with a colon ':' rather than a vertical bar.;
conditional probability: P(X | Y), read "the probability of X given Y".;
divisibility: a | b, read "a divides b".;
the Sheffer stroke in logic: a | b, read "a nand b".;
evaluate: a + 3 | a = 4, read "a plus 3 evaluated for when a equals 4", or 4 + 3 = 7.;
Backus-Naur form
In Backus-Naur form the expression consists of sequences of symbols and/or sequences separated by '|', indicating a choice, the whole being a possible substitution for the symbol on the left.
Read more at Wikipedia.org
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