begin{minipage}[]{0.5textwidth}
    begin{flushleft}
        {large textit{Author}:vspace*{0.5cm} \
        Me}
    end{flushleft}
end{minipage}%
%
begin{minipage}[]{0.5textwidth}
    begin{flushright}
        {large textit{Supervisor}:vspace*{0.5cm} \
        Him}\
        vspace*{0.5cm}
        {large textit{Advisor}:vspace*{0.5cm} \
        Other Guy}
    end{flushright}
end{minipage}%

documentclass{article}
begin{document}
  begin{minipage}[t]{0.5textwidth}
    begin{flushleft}
        large textit{Author}:vspace*{0.5cm} \
        Me
    end{flushleft}
end{minipage}%
%
begin{minipage}[t]{0.5textwidth}
    begin{flushright}
        large textit{Supervisor}:vspace*{0.5cm} \
        Him\
        vspace*{0.5cm}
        large textit{Advisor}:vspace*{0.5cm} \
        Other Guy
    end{flushright}
end{minipage}%
end{document}

documentclass{article}
usepackage[utf8]{inputenc}
usepackage{fancybox}
usepackage{geometry}
thispagestyle{empty}
begin{document}
centering
With verb|vspace{0pt}| at the top of each minipage of this example of a pair of minipages see below: %
\
fbox{
        begin{minipage}{0.8linewidth}vspace{0pt}%
        fontsize{10}{6}selectfont Reference.
    end{minipage}%
    }%
        fbox{%
        begin{minipage}{0.1linewidth}vspace{0pt}%
        fontsize{10}{6}selectfont Reference.
            end{minipage}
    }
        end{document}