I'm trying to figure out how to get the odd and pages to display the current section name and the odd pages to display the title of the paper.

How do I accomplish this?

I'm using article and I have fancyhdr loaded.

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Here's one possibility:

\documentclass[twoside]{article}
\usepackage{fancyhdr}
\usepackage{lipsum}% just to generate text for the example

\fancyhf{}
\pagestyle{fancy}

\title{Some Title}
\author{Some Author}

\makeatletter
\let\hdrtitle\@title
\makeatother

\begin{document}
\section{Test Section One}
\lipsum[1-10]
\section{Test Section Two}
\lipsum[1-10]
\end{document}
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I noticed that the pages shift left and right (as if written for a book). How do I remove that feature? – AlanH Mar 31 '13 at 3:15
@AlanH I am not sure what do you mean with "pages shift left and right". Perhaps that inner/outer margins are different for even and odd pages? If this is so, then you could load the geometry package with the centering option: \usepackage[centering]{geometry}. If you meant other thing, please explain in a little more detail. – Gonzalo Medina Mar 31 '13 at 4:17
When I use your code, the section number will display. It's quite confusing because the section number and the page number are both in the header. How do I just display the section title without the numbering? – AlanH Mar 31 '13 at 8:12
@AlanH just now I saw your comment; I see you opened a new question, with this issue. I've provided an answer to this new question. – Gonzalo Medina Mar 31 '13 at 13:57

titleps is an intuitive alternative to fancyhdr which provides \*title macros the contain the sectional unit titles (in contrast to using article's \leftmark, that has no immediate context).

Pilfering an example structure from Gonzalo:

\documentclass[twoside]{article}
\usepackage{titleps,lipsum}% http://ctan.org/pkg/{titleps,lipsum}

\title{Some Title}
\author{Some Author}

\makeatletter
\let\hdrtitle\@title% To store the title after using \maketitle
\newpagestyle{main}{%
{\hdrtitle}{}{\thepage}} % odd
\makeatother
\pagestyle{main}

\begin{document}
\maketitle
\section{A section}
\lipsum[1-10]
\section{Another section}
\lipsum[1-10]
\end{document}

The main driver for setting headers/footers is given by