Fonts you download from the web are generally either TrueType or OpenType fonts, and these cannot be used with pdfLaTeX without being converted, which is a non-trivial task. But with XeLaTeX or LuaLaTeX you can use these fonts easily.
Be warned, however, that very simple fonts like the Alice font you link in your question may have no boldface or italics, for example, so they should probably not be used for a main document font. The following example works, just by installing the Alice font in whatever location your system stores fonts (this depends on your OS).
Compile it with XeLaTeX, and make sure your source file is encoded as UTF-8.
\documentclass{article}
\usepackage{fontspec}
% If you want to use the font for a small section:
\newfontfamily\alice{Alice}
% If you want to use the font for the whole document (not recommended for this font)
%\setmainfont{Alice}
\begin{document}
{\alice This is in the Alice font.\textbf{Notice that there is no boldface} \textit{or italic.}}
\end{document}

It's possible with fontspec and XeLaTeX to fake bold and italics. This doesn't produce very pretty output, and so I don't really recommend it:
\documentclass{article}
\usepackage{fontspec}
% If you want to use the font for a small section:
\newfontfamily\alice[AutoFakeBold=.7,AutoFakeSlant=.3]{Alice}
% Adjust the numbers to get the look you like (bigger numbers = thicker bold/more slanted italic)
% If you want to use the font for the whole document (not recommended for this font)
%\setmainfont{Alice}
\begin{document}
{\alice This is in the Alice font. \textbf{This is faked bold} \textit{and faked italic.}}
\end{document}

fontspec. – Johannes_B Aug 05 '16 at 18:03\documentclass{...}and ending with\end{document}. – DG' Aug 05 '16 at 19:34fontspecpackage. See the XeTeX part of the question you linked too. Also Using XeLaTeX instead of pdfLaTeX – Alan Munn Aug 05 '16 at 21:40