Several errors here.
[12pt] should go between \documentclass and {article}.
\def\bs{\ensuremath\mathbf} isn't a “shorthand of \boldsymbol, but a wrong definition under many respects. Try it outside of math mode to see. Moreover, \boldsymbol{X} typesets a bold italic X, whereas \mathbf prints it bold upright.
There should be no blank line before any math display environment.
For a single equation, use equation (numbered) or equation* (unnumbered).
There should be no blank line between the end of a display environment and \end{thm}.
Now the bigger problem: \text{Var} is wrong, because in an italic context will print its argument in italics. You want \operatorname{Var} instead or \Var defined by
\DeclareMathOperator{\Var}{Var}
Full fixed example. I used \rv (random variable) instead of a more generic name; using the right semantics is helpful. I also loaded bm for better math boldfacing.
\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newcommand{\rv}[1]{\boldsymbol{#1}} % random variable
% or \mathbf{#1} if you want upright type
\DeclareMathOperator{\Var}{Var}
\begin{document}
\begin{thm}
This is a theorem.
\begin{equation*}
\Var(\rv{A}) = \Var(\rv{Y} + \rv{X})
\end{equation*}
\end{thm}
I would like to have the following in the theorem
\begin{equation*}
\Var(\rv{A}) = \Var(\rv{Y} + \rv{X})
\end{equation*}
\end{document}

\mathrminstead of\text. Or better, use\DeclareMathOperator{\Var}{Var}then use\Var... – Phelype Oleinik Oct 15 '18 at 16:57\textcompletely wrong here. Normal users should only use\textfor textual comments in displayed math. These are functions not comments, mark them as such. – daleif Oct 15 '18 at 17:03\textis very common. May I ask, where do you have it from? – daleif Oct 15 '18 at 17:04\bsis definitely not a shorthand for\boldsymboland, as defined, works by pure chance. – egreg Oct 15 '18 at 17:22