The problem you are running in to is that the :
is treated as a relational operator and thus with white space around it. The white space after a relational operator disappears if followed by the &
-command of the align
environment. You will see this problem a lot on this page when seeing equation which are aligned to the right of their =
-sign. In order to protect the white space, you would have to type H_0:{}&\text{}
or H_0&:\text{}
here.
However, it seems as if you do not want to have a relational operator here. There are three possible ways: The relational operator :
, the "maps to" symbol \colon
or the 'normal' text colon \text{:}
. The following shows the differences:
\begin{align*}
&H_0:x\\
&H_0\colon x\\
&H_0\text{: }x
\end{align*}

You can see the different spacings here. And you may have noticed that the text-colon needs to be followed by a white-space, as there is no space between normal symbols (not operators) in math-mode.
In the following, I will show an MWE with two possible solutions of your desired alignment. I will use the two cases I believe you would like to have.
% arara: pdflatex
\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{align*}
H_0\text{: } &\text{Something.}\\
H_1\text{: } &\text{Something.}
\end{align*}
% or identically
%\begin{align*}
% H_0&\text{: Something.}\\
% H_1&\text{: Something.}
%\end{align*}
\end{document}

\begin{align*}
H_0\colon &\text{Something.}\\
H_1\colon &\text{Something.}
\end{align*}
% or identically
%\begin{align*}
% H_0&\colon \text{Something.}\\
% H_1&\colon \text{Something.}
%\end{align*}

tabular
and putting$H_0$
,$H_1$
inside it.