\mid, | (vertical bar), \vert, \lvert, \rvert, \divides

Another option which was not mentioned in any of the comments above is:

F=\left.\frac{\partial f}{\partial x}\right|_{\hat x_{k-1}}

I hope this helps someone else.

This is similar in spirit to qbi's answer. Let me quote from the guide to the amsmath package (the document known as amsldoc), section 4.14.2 Vertical bar notations:

The amsmath package provides commands \lvert, \rvert, \lVert, \rVert (compare \langle, \rangle) to address the problem of overloading for the vert bar character |. This character is currently used in LaTeX documents to represent a wide variety of mathematical objects [...]. The multiplicity of uses in itself is not so bad; what is bad, however, is that fact that not all of the uses take the same typographical treatment, and that the complex discriminatory powers of a knowledgeable reader cannot be replicated in computer processing of mathematical documents. It is recommended therefore that there should be a one-to-one correspondence in any given document between the vert bar character | and a selected mathematical notation, and similarly for the double-bar command \|. This immediately rules out the use of | and \| for delimiters, because left and right delimiters are distinct usages that do not relate in the same way to adjacent symbols; recommended practice is therefore to define suitable commands in the document preamble for any paired-delimiter use of vert bar symbols:

\providecommand{\abs}[1]{\lvert#1\rvert} \providecommand{\norm}[1]{\lVert#1\rVert}

whereupon the document would contain \abs{z} to produce |z| and \norm{v} to produce ∥v∥.

As far as I know, by default they're all defined to be the same bar, except for maybe some spacing differences. But LaTeX, like properly written HTML, favors semantic markup over purely functional markup - you use the command for what you mean, rather than just what you want to appear on the page. This way, if you decide later on that you want certain kinds of bars to look different, it's easier to change only the bars you actually want to and not mess with anything else. For example, if you want to have less space between the bars and the text in constructions like |x|, you can redefine \lvert and \rvert appropriately.

As qbi said, it is recommended to define a higher level of semantic markup, namely things like \abs, \norm, \union, \or, \suchthat, etc., to represent what you really mean in your formulas, and to use those instead of \vert, \lvert and \rvert directly.

\mid is a relation symbol and | is a delimiter. As far as I know \vert is basically the same as |. For things like abslute value and norm I like to use mathtools.sty. This class allows to define something like \absval{} which translates to \lvert ...\rvert. This is useful when you tend to forget the closing bar. :-)

Old question, but it got bumped to the front page…

The character printed by |, \mid, \lvert and \rvert is always the same, but with different math class:

• | is an ordinary symbol;
• \mid is a relation symbol;
• \lvert is an opening symbol;
• \rvert is a closing symbol.

Access to the last two requires loading amsmath. Except \mid they can all be used in the context of delimiters. For instance, if you need a taller bar as a relation symbol, you can use \bigm|; an example usage could be in

\lvert a\rvert \bigm| b

because \bigm\mid results in an error.

I'd deem a|b to denote divisibility a wrong usage on mathematical grounds, because this is a relation. Personally I always use \mid for this case, but I acknowledge that people might find the spacing excessive.

\documentclass{article}
\usepackage{amsmath}

\newcommand{\divides}{%
  \mathrel{%
    \nonscript\mspace{-\thickmuskip}%
    \nonscript\mspace{0.5\thinmuskip}%
    |
    \nonscript\mspace{0.5\thinmuskip}%
    \nonscript\mspace{-\thickmuskip}%
  }%
}

\begin{document}

$a\mid b$ $A_{a\mid b}$ (with \verb+\mid+)

$a\divides b$ $A_{a\divides b}$ (with \verb+\divides+)

\end{document}

The \nonscript bits are for avoiding the spaces (positive or negative) to be inserted when TeX wouldn't.

I'd endorse using a semantic command for divisibility, even if it is just

\newcommand{\divides}{\mid}

because this will allow change one's mind at any time (which is what I do for bigger projects).

A possible improvement is coping with the \big problem above:

\documentclass{article}
\usepackage{amsmath}

\NewDocumentCommand{\divides}{s}{%
  \mathrel{%
    \nonscript\mspace{-\thickmuskip}%
    \nonscript\mspace{0.5\thinmuskip}%
    \IfBooleanT{#1}{\big}|
    \nonscript\mspace{0.5\thinmuskip}%
    \nonscript\mspace{-\thickmuskip}%
  }%
}

\begin{document}

$a\mid b$ $A_{a\mid b}$ (with \verb+\mid+)

$a\divides b$ $A_{a\divides b}$ (with \verb+\divides+)

$\lvert a\rvert\divides* b$

\end{document}

I'd not generalize to other sizes, because of the symbol's meaning.

In general | should be avoided for the absolute value: compare

$|-1+x|=|x-1|$

$\lvert -1+x\rvert=\lvert x-1\rvert$

The top one is definitely wrong: since | is ordinary, the minus sign finds two ordinary symbols around it and so it typesets as a binary operatio symbol; with the correct syntax, \lvert is an opening, so the minus sign stays unary as it should.

I think it is helpful to compare the definitions of these commands:

%%% From kernel

% \mid
\DeclareMathSymbol{\mid}{\mathrel}{symbols}{"6A}
% | and \vert (the same)
\DeclareMathDelimiter{|}{\mathord}{symbols}{"6A}{largesymbols}{"0C}
\DeclareMathDelimiter{\vert}{\mathord}{symbols}{"6A}{largesymbols}{"0C}
% \| and \Vert (the same)
\DeclareMathDelimiter{\Vert}
   {\mathord}{symbols}{"6B}{largesymbols}{"0D}
\DeclareMathDelimiter{\|}
   {\mathord}{symbols}{"6B}{largesymbols}{"0D}

%%% From amsmath

% \lvert and \rvert
\DeclareMathDelimiter{\lvert}
  {\mathopen}{symbols}{"6A}{largesymbols}{"0C}
\DeclareMathDelimiter{\rvert}
  {\mathclose}{symbols}{"6A}{largesymbols}{"0C}
% \lVert and \rVert
\DeclareMathDelimiter{\lVert}
  {\mathopen}{symbols}{"6B}{largesymbols}{"0D}
\DeclareMathDelimiter{\rVert}
  {\mathclose}{symbols}{"6B}{largesymbols}{"0D}

So we summarize as follows:

The \mid is special since it's the only relational symbol here. And we should use it in some cases:

1. In division notation: a\mid b
2. In set representation: \{ x \mid x\neq 0 \}

Another option is instead of using

\left. ... \right|

one can use

F=\frac{\partial f}{\partial x}\bigg|_{\hat x_{k-1}}