There are a couple things you could try. A simple technique is to use \hfill, such as
\documentclass{article}
\begin{document}
\textbf{\underline{Student:}}
\hfill
\textbf{\underline{Supervisor:}} \\
Mr. Thatshisname
\hfill
Prof. Whatshisname
\end{document}
This will produce the following:

This does not quite reproduce the spacing for the "Supervisor" word as shown in your sample. Another approach will reproduce this better, although it's a little more involved. This uses \hfill along with the minipage environment:
\documentclass{article}
\begin{document}
\begin{minipage}{2in}
\textbf{\underline{Student:}} \\
Mr. Thatshisname
\end{minipage}
\hfill
\begin{minipage}{1.3in}
\textbf{\underline{Supervisor:}} \\
Prof. Whatshisname
\end{minipage}
\end{document}
The size of the first minipage does not matter too much, since \hfill will fill the space, but the size of the second minipage needs to be tweaked to get the right edge as close to the margin as possible without forcing a line break. I used the showframe package to make that process easier, but perhaps someone else has a more elegant solution. Here is the output:

In order to get this on the bottom of the page, you can try using the \vfill command. Another approach to achieving the result you're looking for is by using the fancyhdr package (read "fancy header"):
\documentclass{article}
\usepackage{fancyhdr}
\renewcommand{\headrulewidth}{0pt}
\usepackage{lipsum}
\pagestyle{fancy}
\lfoot{\textbf{\underline{Student:}} \\
Mr. Thatshisname}
\rfoot{\textbf{\underline{Supervisor:}}
\phantom{MMMi}\\
Prof. Whatshisname}
\begin{document}
\lipsum[1-4]
\end{document}
This will place these words on the bottom of every page with the pagestyle fancy. If you want it only on the first page, then use \thispagestyle{fancy} after \begin{document} instead of \pagestyle{fancy} as shown in my example. I had to redefine the \headrulewidth to zero so that the line did not show up at the top of the page (can someone comment on this answer if there is a "nicer" way to do this?). This solution also has the same problem as the first, where "Supervisor" is pushed all the way to the margin. However, I used the \phantom command to add some white space, and fiddled with the input to make it look right. Here is that output:

Hope this helps!
hfilltriggers an 'underfull hbox' alert. How should I fix it? – Fifnmar Mar 25 '23 at 05:52