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The following code

documentclass{article}





usepackage{amsthm}

usepackage{amsmath}

usepackage{mathtools}



usepackage[left=1.5in, right=1.5in, top=0.5in]{geometry}







newtheorem{definition}{Definition}

newtheorem{theorem}{Theorem}





begin{document}

    title{Extra Credit}

    maketitle



    begin{definition}

        If f is analytic at $z_0$, then the series



        begin{equation}

            f(z_0) + f'(z_0)(z-z_0) + frac{f''(z_0)}{2!}(z-z_0)^2 + cdots = sum_{n=0}^{infty} frac{f^{(n)}(z_0)}{n!}(z-z_0)^n

        end{equation}



        is called the Taylor series for f around $z_0$.

    end{definition}



    begin{theorem}

        If f is analytic inside and on the simple closed positively oriented contour $Gamma$ and if $z_0$ is any point inside $Gamma$, then

        begin{equation}

            f^{(n)}(z_0) = frac{n!}{2pi i} int_{Gamma} frac{f(zeta)}{(zeta - z_0)^{n+1}}dzeta hspace{1cm} (n=1,2,3, cdots )

        end{equation}

    end{theorem}



begin{theorem}

(Cauchy's Integral Formula) Let $Gamma$ be a simple closed positively oriented contour. If $f$ is analytic in some simply connected domain $D$ containing $Gamma$ and $z_0$ is any point inside $Gamma$, then

begin{equation}

f(z_0)= frac{1}{2pi i} int_{Gamma} frac{f(z)}{z-z_0} dz

end{equation}

end{theorem}

noindent hrulefill



begin{theorem}

If f is analytic in the disk $|z-z_0|<R'$, then the Taylor series $(1)$ converges to $f(z)$ for all $z$ in this disk. 

end{theorem}

produces the following image

How can I enclose Definition 1, Theorem 1, and Theorem 2 in separate rectangles. And have these rectangles separated by a space?

You can try with shadethm package, it can do all you want and many more. In you example what you need is:

documentclass{article}

usepackage{shadethm}

usepackage{mathtools}



newshadetheorem{boxdef}{Definition}[section]

newshadetheorem{boxtheorem}[boxdef]{Theorem}

newtheorem{theorem}[boxdef]{Theorem}



setlength{shadeboxsep}{2pt}  

setlength{shadeboxrule}{.4pt}  

setlength{shadedtextwidth}{textwidth}

addtolength{shadedtextwidth}{-2shadeboxsep}

addtolength{shadedtextwidth}{-2shadeboxrule}

setlength{shadeleftshift}{0pt} 

setlength{shaderightshift}{0pt}

definecolor{shadethmcolor}{cmyk}{0,0,0,0}

definecolor{shaderulecolor}{cmyk}{0,0,0,1}



begin{document}



section{Boxed theorems}



begin{boxdef}

        If f is analytic at $z_0$, then the series



        begin{equation}

            f(z_0) + f'(z_0)(z-z_0) + frac{f''(z_0)}{2!}(z-z_0)^2 + cdots = sum_{n=0}^{infty} frac{f^{(n)}(z_0)}{n!}(z-z_0)^n

        end{equation}



        is called the Taylor series for f around $z_0$.

end{boxdef}



begin{boxtheorem}

        If f is analytic inside and on the simple closed positively oriented contour $Gamma$ and if $z_0$ is any point inside $Gamma$, then

        begin{equation}

            f^{(n)}(z_0) = frac{n!}{2pi i} int_{Gamma} frac{f(zeta)}{(zeta - z_0)^{n+1}}dzeta hspace{1cm} (n=1,2,3, cdots )

        end{equation}

end{boxtheorem}



begin{boxtheorem}

(Cauchy's Integral Formula) Let $Gamma$ be a simple closed positively oriented contour. If $f$ is analytic in some simply connected domain $D$ containing $Gamma$ and $z_0$ is any point inside $Gamma$, then

begin{equation}

f(z_0)= frac{1}{2pi i} int_{Gamma} frac{f(z)}{z-z_0} dz

end{equation}

end{boxtheorem}

noindent hrulefill



begin{theorem}

If f is analytic in the disk $|z-z_0|<R'$, then the Taylor series $(1)$ converges to $f(z)$ for all $z$ in this disk. 

end{theorem}



end{document}

which produces the following:

Here is a solution with thmtools, which cooperates wit amsthm. Unrelated: you don't have to load amsmath if you load mathtools, as the latter does it for you:

documentclass{article}

usepackage{amsthm, thmtools}

usepackage{mathtools}



usepackage[left=1.5in, right=1.5in, top=0.5in]{geometry}



newtheorem{definition}{Definition}

newtheorem{theorem}{Theorem}



declaretheorem[sibling=definition, shaded={rulecolor=black, rulewidth=0.6pt, bgcolor={rgb}{1,1,1}},name=Definition]{boxeddef}

declaretheorem[sibling=theorem, shaded={rulecolor=black, rulewidth=0.6pt, bgcolor={rgb}{1,1,1}},name=Theorem]{boxedthm}



begin{document}

title{Extra Credit}

author{}

maketitle



begin{boxeddef}

    If f is analytic at $z_0$, then the series



    begin{equation}

        f(z_0) + f'(z_0)(z-z_0) + frac{f''(z_0)}{2!}(z-z_0)^2 + cdots = sum_{n=0}^{infty} frac{f^{(n)}(z_0)}{n!}(z-z_0)^n

    end{equation}



    is called the Taylor series for f around $z_0$.

end{boxeddef}



begin{boxedthm}

    If f is analytic inside and on the simple closed positively oriented contour $Gamma$ and if $z_0$ is any point inside $Gamma$, then

    begin{equation}

        f^{(n)}(z_0) = frac{n!}{2pi i} int_{Gamma} frac{f(zeta)}{(zeta - z_0)^{n+1}}dzeta hspace{1cm} (n=1,2,3, cdots )

    end{equation}

end{boxedthm}



begin{boxedthm}

(Cauchy's Integral Formula) Let $Gamma$ be a simple closed positively oriented contour. If $f$ is analytic in some simply connected domain $D$ containing $Gamma$ and $z_0$ is any point inside $Gamma$, then

begin{equation}

f(z_0)= frac{1}{2pi i} int_{Gamma} frac{f(z)}{z-z_0} dz

end{equation}

end{boxedthm}



noindent hrulefill



begin{theorem}

If f is analytic in the disk $|z-z_0|<R'$, then the Taylor series $(1)$ converges to $f(z)$ for all $z$ in this disk.

end{theorem}



end{document}