Convert to Xelatex

Here's (real) minimal example that solves your problem.

\documentclass{article}

\usepackage{fontspec}
%\usepackage{mathptmx} % don't use this package with XeLaTeX
\setmainfont{TeX Gyre Termes}
\begin{document}
\textbf{Some text}
\end{document}

Independently, don't pass the xelatex option to hyperref.

You get

LaTeX Font Warning: Font shape `TU/ptm/m/n' undefined
(Font)              using `TU/lmr/m/n' instead on input line 23.

LaTeX Font Warning: Font shape `TU/ptm/bx/n' undefined
(Font)              using `TU/ptm/m/n' instead on input line 25.

and this means you won't get Times, but Latin Modern, with no font shape or series variation.

The error is loading mathptmx. If you want Times, there are several choices, depending on your setup. If you just have BasicTeX, forget it: it doesn't have most of the useful things. However Mac OS X provides an OpenType Times font.

I fixed several errors; mainly package loading and names of environments: don't use \begin{raggedleft} or \begin{raggedright}, but \begin{flushright} and \begin{flushleft} respectively.

See also the comments I added.

\documentclass[letterpaper,11pt]{article}

\usepackage[margin=1.905cm]{geometry}
\usepackage{fontspec}
\usepackage{ulem}

%\usepackage[none]{hyphenat} % Loading this is a recipe for bad typography
\usepackage{enumitem}
\usepackage[hyphens]{url}
\urlstyle{same}
\usepackage{hyperref} % no xelatex nor xetex option; it should be loaded last
\hypersetup{
    colorlinks=true,
    urlcolor=blue,
    }

%\pagenumbering{gobble} % never ever use this, particularly with hyperref
\pagestyle{empty} % this is good

\setmainfont{Times New Roman}

\newcommand\myauthor{Christopher M. Finazzo} % not \def
\newcommand\mycopyright{2018}

\begin{document}

\begin{center}
\bfseries\small CHRISTOPHER M. FINAZZO
\end{center}

\begin{flushright}
973.906.0496\\
\end{flushright}

\begin{flushleft}
[email protected]\\
http://www.linkedin.com/in/chrisfinazzo
\end{flushleft}

\begin{center}
\textbf{PROFESSIONAL OBJECTIVE} \\
Communications Analyst
\end{center}

\begin{center}
\textbf{PREFERRED FUNCTIONS} \\
\begin{tabular}[t]{lll}
Marketing Communications & Internal Communications & Client Communications\\
\end{tabular} \\[1ex]
\end{center}

\begin{flushleft}
{\centering\textbf{POSITIONING STATEMENT}\\}
Results driven individual with a passion for conveying complex 
ideas to technical and nontechnical audiences in a way that is 
accurate, clear, and concise. Proven ability to create audience 
appropriate messages with a range of authoring tools. Strong 
verbal and written communication skills as well as the ability 
to multitask, set priorities, and meet deadlines.
\end{flushleft}

\begin{center}
\textbf{COMPETENCIES} \\
\begin{tabular}[t]{lll}
\textbf{Writing \& Editing} & \textbf{Authoring Tools} & \textbf{Project Management}\\
AP \& APA Style & HTML, CSS, \& SASS & Requirements Gathering\\
Research Skills & Content Management Systems & Estimating Project Scope\\
Creating a Written Narrative & Social Media & Audience Analysis\\
\end{tabular}
\end{center}

\begin{flushleft}
{\centering\textbf{TARGET MARKET CHARACTERISTICS}\\}
\textbf{Geographic Location:} New York Metro area, Should be commutable via public transit.\\
\textbf{Industry or Type of Organization:} Communications, Financial Services, Insurance, Technology\\
\textbf{Size of Organization:} Medium to Large\\
\textbf{Organizational Culture:}\\
Values employees\\
Encourages employee autonomy and individual initiative\\
Provides opportunities for personal and professional growth\\
Leveraging technology to engage with target audiences
\end{flushleft}

\begin{center}
\textbf{TARGET LIST} \\
\begin{tabular}[t]{ll}
\textbf{Financial Services/Insurance} & \textbf{Communications/Tech/Other}\\
ACE Insurance Group & Bloomberg\\
AIG & Columbia University\\
American Express & Coyne Public Relations\\
Bank of America & Discovery Communications\\
BlackRock & Edelman\\
BNY Mellon & Facebook\\
Chubb & Google\\
Citigroup & HP\\
Goldman Sachs & IBM\\
JPMorgan Chase & Microsoft\\
Liberty Mutual & The New York Times\\
Morgan Stanley & NYU\\
New York Life & Rapp\\
Prudential & PwC\\
Travelers & The Star Ledger\\
UBS & Unilever\\
\end{tabular}
\end{center}

\end{document}

The typesetting is very unbalanced, though.