After troubleshooting with @Wyck in chat, we have narrowed to the underlying issue.
The issue was related to my server configuration not having the proper amount of memory allocated to Apache/PHP.
If anyone has this same problem, please try verifying that you have enough (64 MB+) server memory allocated to Apache/PHP in your server configuration settings. You can also add this to your wp-config.php file:
If the above solution does not work, read the article Image/Media Uploader problems? for further troubleshooting.
Try adding one by one or all of the following .htaccess tweaks to the .htaccess file in the root directory of your WordPress installation.
Seriously, try one of each of the solutions below so you know which one did the trick. Don't just paste them all in your .htaccess file immediately.
Try this line:
AddType x-mapp-php5 .php
mod_security might be causing problems. Disable it to see if that is the problem. To do this, make an .htaccess file in your wp-admin directory. Add this to it:
If you're using access control based on authentication on your Webserver (often known as htpasswd, basic authentication, password protected directory or similar), WordPress is not able to handle it for Flash Uploader, Cron, and XMLRPC. Related files need to be excluded to work. Keep in mind that this might break your security considerations.
# Exclude the file upload and WordPress CRON scripts from authentication
Allow from all
Deny from none
One final note: some have said that if using a lesser version of PHP 5.3.X you can try disabling PHP Safe Mode.
If you are running WordPress multi-site and are receiving HTTP errors or internal server errors, related to image uploading, please read Uploading Images to Multi-Site Causes Failure to HTTP Error) for other possible troubleshooting ideas and solutions.