Option 1: LaTeX
Select the Times font, which looks a lot like Times New Roman. More info here.
MWE
\documentclass{article}
\begin{document}
My cat has black skin. % in primary font
{\fontfamily{ptm}\selectfont
My second cat has white skin.} % I would like to get in Times New Roman
\end{document}
Result

Option 2: XeLaTeX/LuaLaTeX
Select fonts installed on you computer using fontspec as Johannes_B suggested. Now you have to compile using XeLaTeX or LuaLaTeX though. (thanks clemens)
MWE
\documentclass{article}
\usepackage{fontspec}
% Compile with XeLaTeX
\newfontfamily\myfont{times.ttf}
\begin{document}
My cat has black skin. % in primary font
{\myfont
My second cat has white skin.} % I would like to get in Times New Roman
\end{document}
Result

fontspec. – Johannes_B Sep 24 '16 at 07:24fontspec. – Johannes_B Sep 24 '16 at 07:29