How to check if a value is not equal using TeX conditionals?

You can define a command that expands to "apple", another that expands to whatever you want to test, and then use \ifx. Here's a latex file that demonstrates this:

\documentclass{article}

\begin{document}

\def\appleref{apple}
\def\testit#1{%
  \def\temp{#1}%
  \ifx\temp\appleref
    Yes, it's apple.
  \else
    No, it's #1.
  \fi
}


apple: \testit{apple}

pear: \testit{pear}

\def\fakeapple{apple}

fakeapple: \testit{\fakeapple}

\end{document}

Edit: tohecz points out in a comment that if you change \def\temp{#1} to \edef\temp{#1}, then \fakeapple (which is a macro that expands to "apple") would test as being equal to "apple".

ConTeXt provides a \doif... series of macros to do string comparisons. See the ConTeXt wiki for details. For example, if you want check if #1 is the same as a previously defined macro \fakeapple, then you can use:

\def\checkapple#1%
    {\doifnot\fakeapple{#1}
       {It is not an apple, it is #1}}

Under pdfTeX you can perform string comparisons using \pdfstrcmp{<strA>}{<strB>}. From the pdfTeX user manual:

\pdfstrcmp{<general text>}{<general text>} (expandable)

This command compares two strings and expands to 0 if the strings are equal, to -1 if the first string ranks before the second, and to 1 otherwise. The primitive was introduced in pdfTEX 1.30.0.

\documentclass{article}

\begin{document}

\def\appleref{apple}
\def\testit#1{%
  \ifnum\pdfstrcmp{#1}{apple}=0
    Yes, it's apple.
  \else
    No, it's #1.
  \fi
}

apple: \testit{apple}

pear: \testit{pear}

\def\fakeapple{apple}

fakeapple: \testit{\fakeapple}

\end{document}