I'm attempting to learn how to incorporate Google Analytics and custom event tracking on a new website. After researching into it, I can't seem to understand the difference and relationship between Google Tag Manager and just writing custom event tracking code.
Is Google Tag Manager just an easier way to track information without having to write JS?
Yes - Google Tag Manager is a "Tag Management System" wherein you can implement a lot of tags external of a typical development release cycle.
Adding tags without efficient management prompts big problems. All that code bogs down your site. Redundant or incorrectly applied tags can distort your measurement and result in duplicate costs or missing data. And it can be time-consuming for the IT department or webmaster team to add new tags, which means important marketing and measurement programs can be delayed. With Google Tag Manager, you eliminate these problems—and run your campaigns when you need them. -- source
Marketing and tracking services provide “tags” — snippets of code you’re often asked to embed right before the end of your tag — and they’re relatively easy to use, but when more than a few tags are involved, managing them can be a tedious task. This is where Google’s new Tag Manager comes in, which consolidates all of your site’s tags into a single admin panel. Now, you only have to include one automatically-updated snippet on your site. -- source
A Tag Management System (TMS) is solution that controls the deployment of externally hosted Javascript "page tags", very similar to what a Content Management System does for content. Probably the most known tag-based application known is Google Analytics, but other apps include simple social media interactions such as the Facebook Like or Tweet This buttons to more complex systems such as the Intense Debate or Disqus comments systems. -- source
So in your question, you can implement google analytics via custom event tracking code or you can install google tag manager and then code all your custom google analytics event hooks in it.