The commonest hand-written style for expressions is to place the limit
expressions on operators such as \sum and \int physically
above and below the operator. In (La)TeX, we write these limit
expressions using sub- and superscripts applied to the operator, but
they don’t always appear in the “handwritten” way in TeX’s
output.
The reason is, that when an expression appears in non-display maths,
in running text (and is therefore in TeX \textstyle), placing
the limits thus could lead to ragged line spacing (and hence
difficult-to-read text). It is therefore common (in \textstyle)
to place the limits as one would sub- and superscripts of variables.
This is not universally satisfactory, so the primitive \limits is
provided:
$\sum\limits_{n=1}^{m} ...$
which will place the limits right above and below the symbol (and be
blowed to the typography…).
Contrariwise, you may wish to change the arrangement of the limits
when in \displaystyle. For this purpose, there’s a corresponding
\nolimits:
\[\sum\nolimits_{n=1}^{m} ...\]
which will place the limits as they would be in \textstyle.
Alternatively, one can manipulate the
\textstyle/\displaystyle state of the mathematics. To get
“\limits placement” in inline maths,
$\displaystyle\sum_{n=1}^{m} ...$
and for “\nolimits placement” in display maths,
\nolimits:
\[\textstyle\sum_{n=1}^{m} ...\]
will serve. Either of these forms may have effects other than on the
operator you’re considering, but there are still those who prefer this
formulation.
Remember, if you’re declaring a special operator of your own, the AMSLaTeX functions (that you ought to be using) allow you to choose how limits are displayed, at definition time.
(Note that the macro \int normally has \nolimits built in to
its definition. There is an example in the TeXbook to show how odd
\int\limits looks when typeset.)
This question on the Web: http://www.tex.ac.uk/cgi-bin/texfaq2html?label=limits