\hspace is not working after \newline command

What your code does

From the unofficial LaTeX reference manual, sec. 20.1:

LaTeX normally removes horizontal space that comes at the beginning or end of a line. To preserve this space, use the optional * form [of \hspace].

So let’s use \hspace* instead:

\documentclass[]{article}
\begin{document}
    \section*{Affirmation}
    \addcontentsline{toc}{section}{Affirmation}
    \hspace{1cm}something
    \vspace{2cm}
    \hspace{1cm}city, date \hspace{3cm}\line(1,0){200}
    \newline
    \hspace*{8cm}(name)
    \clearpage
\end{document}

What your picture says

The rest of the spacing is pretty different from your output because your screen shot was quite certainly not produced by the code you posted. With a bit of guesswork, I think this is closer to what you intended (note that you have to use the starred version of \hspace in several instances):

\documentclass{article}
\begin{document}
\section*{Affirmation}
\addcontentsline{toc}{section}{Affirmation}
\hspace*{1cm}something

\vspace{2cm}

\noindent\hspace*{1cm}city, date \hspace{3cm}\line(1,0){200}
\newline
\noindent\hspace*{8cm}(name)
\end{document}

What I would suggest

Taking your layout and thinking about an implementation, I would do something like this, which is cleaner than meddling with a lot of spacing manually:

\documentclass{article}
\begin{document}
\section*{Affirmation}
\addcontentsline{toc}{section}{Affirmation}

something\\[2cm]
city, date \hfill
\begin{minipage}[t]{7cm}\centering
    \rule{\linewidth}{.5pt}\\
    (name)
\end{minipage}

\end{document}

As @doncherry said you can fix the problem using the starred version \hspace*{}. But I'm giving here an alternative solution to you to produce a line to sign. I usually use it.

\begin{flushright}
\begin{minipage}{6cm}
\begin{center}
\hrulefill\\
(date)
\end{center}
\end{minipage}
\end{flushright}