if test for 1 not 11

Here is an \ifnum comparison that can handle non-numeric inputs. It does it by comparing 1 with 0#1. If #1 is a number, the \ifnum comparison works in the standard way, since the leading 0 does not affect the numerical value.

If #1 is not a number, then the \ifnum argument expansion stops after the 0 with a false comparison (having already printed out #1), and then prints out things.

\documentclass{article}
\newcommand{\sth}[1]{\textbf{(#1 \ifnum1=0#1\relax thing\else things\fi)}}
\begin{document}
\sth{1}  vs. \sth{2} vs. \sth{11} vs. \sth{no}
\end{document}

\pdfstrcmp of pdfTeX expands two token groups and compares them as strings. If they are equal, \pdfstrcmp expands to 0, which can be tested with \ifnum or \if:

\ifnum\pdfstrcmp{#1}{1}=0 % space ends the number zero here
  ...
\else
  ...
\fi

or

\if0\pdfstrcmp{#1}{1}%
  ...
\else
  ...
\fi

Package pdftexcmds' \pdf@strcmp hides the different names and methods of the different TeX engines (pdfTeX, XeTeX, LuaTeX). "Vanilla" TeX is not supported.

If #1 can contain fragile stuff, then the LaTeX way is a protected expansion, e.g.:

\protected@edef\param{#1}%
\def\one{1}%
\ifx\one\param
  ...
\else
  ...
\fi

Or if the argument should not be expanded at all:

\def\param{#1}%
\def\one{1}%
\ifx\param\one
  ...
\else
  ...
\fi