A bit of background first. When Knuth wrote TeX, he realised that most macros would not need to absorb more than one paragraph as an argument. As a result, a good way to test for errors such as a missing } is to forbid macros to accept paragraph tokens (either those generated by blank lines or explicit \par tokens). So he created \def for definitions which cannot take a paragraph token, and the \long prefix to allow them to:
\def\examplea#1{% #1 cannot contain \par
}
\long\def\exampleb#1{% #1 can contain \par
}
When LaTeX was written, Lamport created \newcommand as a wrapper around \def with various bits of error checking. He found that paragraph tokens can pop up in odd places. So he decided that \newcommand would be 'long'. When the LaTeX team took over for LaTeX2e, they decided that you would need to add the modifier * to prevent this:
\newcommand{\examplea}[1]{% #1 can contain \par
}
\newcommand*{\exampleb}[1]{% #1 cannot contain \par
}
Most of the time, \newcommand* is the best choice as you want the error-checking that it provides. That is why examples given by experienced LaTeX users normally use this form, rather than just \newcommand.
The same behaviour is seen with \newenvironment:
\newenvironment{examplea}[1]{% #1 can contain \par
}{}
\newenvironment*{exampleb}[1]{% #1 cannot contain \par
}{}
This works by defining \examplea and \endexamplea more-or-less using \newcommand, and \exampleb and \endexampleb more-or-less using \newcommand*. As a result, the 'end' macros have the same 'long' status as the 'begin' ones, even though they never take arguments. (Note that this does not affect what can go into the body of the environment, only the arguments at the start.) Environments are covered in more detail in What is the difference between \newenvironment and \newenvironment*?.
For LaTeX3, we've decided to take a somewhat hybrid approach. If you use xparse for defining document commands, they are only 'long' if you ask for it:
\NewDocumentCommand\examplea{m}{% #1 cannot contain \par
}
\NewDocumentCommand\examplab{+m}{% #1 can contain \par
}
(We've decided on + to represent a long argument). On the other hand, for internal use it is normally best to accept paragraph tokens, and to leave the filtering to the user interface level.
The star at the end of the name of a displayed math environment causes that the formula lines won't be numbered. Otherwise they would automatically get a number.
You can read about that in the amsmath user's guide since align belongs to amsmath.
Such information can usually be found in the package documentation. Type texdoc packagename at the command prompt or visit http://ctan.org/pkg/packagename. If you use google or another search engine, look for starred and commandname.
Commonly, a star symbol * means a version of a command that behaves differently from the original. That often means suppressing numbering but could also refer to special features. For example, have a look at What's the difference between \newcommand and \newcommand*?
Best Answer
Like with all sectioning commands from
\sectionand lower (in the standard document classes), they are defined in terms of\@startsection, which allows for a starred variant. Taken fromlatex.ltx(with emphasis/comments added):The difference only becomes noticeable when looking at the inner workings of
\@ssectand\@sect- the choices after conditioning on*- the former associated with the starred-version. The main differences include:Numbering, based on the value of the counter
secnumdepth. For\paragraph, this is 4. Settingsecnumdepthto anything higher than or equal to 4 would reveal the difference;Inclusion in the ToC. Regardless of the value of
tocdepth- the counter deciding which sectional units to include in the ToC, the entry is written to the.tocfile. Only upon reading the.tocfile is it decided to include the entry or not, based on the value oftocdepth.The running heads is triggered by
\paragraphmark(when calling\paragraph{..}), initially set to\@gobble. So, without any modifications, this just gobbles its argument - the "paragraph title" and changes nothing in the heading. However, a suitable definition of\paragraphmarkwould show a different effect when using\paragraphas opposed to\paragraph*where no\paragraphmarkis called.