Here are some more specifics:
I'm running php 5.5.12, Moodle 2.7 (build 20140605), mysql 5.5.32.01, linux shared server via Siteground.
The error logs on the site aren't proving to be terribly helpful, as they aren't showing errors relevant to this issue.
I'm not sure if it's that the cron job wasn't running, or if there was another error. I contacted Siteground who reassured me that the cron was functioning with no errors. I have been receiving emails stating the same (see below). What is confusing to me is that there is no information and the cron that has "successfully completed" has nothing inside of it. It's as if it completed it's task of doing nothing.
I used EasyCron to attempt to run cron as well. This has been running successfully, according to the site. However, the error message saying that the cron script hasn't run in 24 hours still appears.
I triple checked all of my PHP settings, and found one error in coding in the php.ini. This led me to believe (I'm a newbie) that a reinstall wasn't the worst idea. Backed it all up, made sure everything was up to date, and performed an upgrade to the latest weekly build.
When going to notifications, I saw the error posted below.