Footnotes
Place a numbered footnote at the bottom of the current page, as here.

Noël Coward quipped that having to read a footnote is like having
to go downstairs to answer the door, while in the midst of making
love.\footnote{I wouldn't know, I don't read footnotes.}

You can place multiple footnotes on a page. If the text becomes too long it will flow to the next page.
You can also produce footnotes by combining the \footnotemark and the \footnotetext commands, which is useful in special circumstances.
To make bibliographic references come out as footnotes you need to include a bibliographic style with that behavior.

\footnote
Synopsis:

\footnote[number]{text}

Place a numbered footnote text at the bottom of the current page.

There are over a thousand footnotes in Gibbon's 
\textit{Decline and Fall of the Roman Empire}.\footnote{After
reading an early version with endnotes David Hume complained, 
``One is also plagued with his Notes, according to the present Method 
of printing the Book'' and suggested that they ``only to be printed 
at the Margin or the Bottom of the Page.''}

The optional argument number allows you to specify the footnote number. If you use this option then the footnote number counter is not incremented, and if you do not use it then the counter is incremented.
Change how LaTeX shows the footnote counter with something like \renewcommand{\thefootnote}{\fnsymbol{footnote}}, which uses a sequence of symbols (see \alph \Alph \arabic \roman \Roman \fnsymbol). To make this change global put that in the preamble. If you make the change local then you may want to reset the counter with \setcounter{footnote}{0}. By default LaTeX uses arabic numbers.
LaTeX’s default puts many restrictions on where you can use a \footnote; for instance, you cannot use it in an argument to a sectioning command such as \chapter (it can only be used in outer paragraph mode). There are some workarounds; see following sections.
In a minipage environment the \footnote command uses the mpfootnote counter instead of the footnote counter, so they are numbered independently. They are shown at the bottom of the environment, not at the bottom of the page. And by default they are shown alphabetically. See minipage.

\footnotemark
Synopsis, one of:

\footnotemark
\footnotemark[number]

Put the current footnote number in the text. (See \footnotetext for giving the text of the footnote separately.) The version with the optional argument number uses that number to determine the mark printed. This command can be used in inner paragraph mode.
This example gives the same institutional affiliation to both the first and third authors (\thanks is a version of footnote).

\title{A Treatise on the Binomial Theorem}
\author{J Moriarty\thanks{University of Leeds} 
  \and A C Doyle\thanks{Durham University} 
  \and S Holmes\footnotemark[1]}
\begin{document}
\maketitle

If you use \footnotemark without the optional argument then it increments the footnote counter but if you use the optional number then it does not. This produces several consecutive footnote markers referring to the same footnote.

The first theorem\footnote{Due to Gauss.} 
and the second theorem\footnotemark[\value{footnote}] 
and the third theorem.\footnotemark[\value{footnote}]

\footnotetext
Synopsis, one of:

\footnotetext{text}
\footnotetext[number]{text}

Place text at the bottom of the page as a footnote. This command can come anywhere after the \footnotemark command. The optional argument number changes the displayed footnote number. The \footnotetext command must appear in outer paragraph mode.

Footnotes in a table
Inside a table environment the \footnote command does not work. For instance, if the code below appears on its own then the footnote simply disappears; there is a footnote mark in the table cell but nothing is set at the bottom of the page.

\begin{center}
     \begin{tabular}{l|l}
     \textsc{Ship}  &\textsc{Book} \\ \hline 
     \textit{HMS Sophie}     &Master and Commander  \\ 
     \textit{HMS Polychrest} &Post Captain  \\  
     \textit{HMS Lively}     &Post Captain \\
     \textit{HMS Surprise}   &A number of books\footnote{Starting with HMS Surprise.}
     \end{tabular}
\end{center}

The solution is to surround the tabular environment with a minipage environment, as here (see minipage).

\begin{center}
  \begin{minipage}{.5\textwidth}
    .. tabular material ..
  \end{minipage}
\end{center}

The same technique will work inside a floating table environment (see table). To get the footnote at the bottom of the page use the tablefootnote package, as illustrated in this example. If you put \usepackage{tablefootnote} in the preamble and use the code shown then the footnote appears at the bottom and is numbered in sequence with other footnotes.

\begin{table}
  \centering
     \begin{tabular}{l|l}
     \textsc{Date}  &\textsc{Campaign} \\ \hline 
     1862           &Fort Donelson \\ 
     1863           &Vicksburg     \\  
     1865           &Army of Northern Virginia\footnote{Ending the war.}
     \end{tabular}
    \caption{Forces captured by US Grant}
\end{table}

Footnotes in section headings
Putting a footnote in a section heading

\section{Full sets\protect\footnote{This material is due to R~Jones.}}

causes the footnote to appear both at the bottom of the page where the section starts and at the bottom of the table of contents page. To have it not appear on the table of contents use the package footmisc with the stable option.

\usepackage[stable]{footmisc}
 ..
\begin{document}
 ..
\section{Full sets\footnote{This material is due to R~Jones.}}

Note that the \protect is gone; putting it in causes the footnote to reappear on the table of contents.

Footnotes of footnotes
Particularly in the humanities, authors can have multiple classes of footnotes, including having footnotes of footnotes. The package bigfoot extends LaTeX’s default footnote mechanism in many ways, including allow these two, as in this example.

\usepackage{bigfoot}
\DeclareNewFootnote{Default}
\DeclareNewFootnote{from}[alph]   % create class \footnotefrom{}
  ..
\begin{document}
  ..
The third theorem is a partial converse of the 
second.\footnotefrom{First noted in Wilson.\footnote{Second edition only.}}
 ..

Multiple references to footnotes
You can refer to a single footnote more than once. This example uses the package cleverref.

\usepackage{cleveref}[2012/02/15]   % this version of package or later 
\crefformat{footnote}{#2\footnotemark[#1]#3}
  ..
\begin{document}
  ..
The theorem is from Evers.\footnote{\label{fn:TE}Tinker and Evers, 1994.}
The corollary is from Chance.\footnote{Evers and Chance, 1990.}
But the key lemma is from Tinker.\cref{fn:TE}
  ..

This solution will work with the package hyperref. See \footnotemark for a simpler solution in the common case of multiple authors with the same affiliation.

Footnote parameters
\footnoterule 	Produces the rule separating the main text on a page from the page’s footnotes. Default dimensions: 0.4pt thick (or wide), and 0.4\columnwidth long in the standard document classes (except slides, where it does not appear).
\footnotesep 	The height of the strut placed at the beginning of the footnote. By default, this is set to the normal strut for \footnotesize fonts (see Font sizes), therefore there is no extra space between footnotes. This is ‘6.65pt’ for ‘10pt’, ‘7.7pt’ for ‘11pt’, and ‘8.4pt’ for ‘12pt’.