Forgive me if there is already a post answering this, but I am new to this and have not the slightest idea on how this works. I checked my account settings and saw that it said I have a premium account until the fourth of April. So, what exactly is a premium account?

See the Help tab -> FAQ, it has all the information about premium membership :)

Make the most of it while you can. Some of the features are very useful, like teleporting off an island at any time (instead of having to use the portal) and automatic warnings about inbreeding (without needing to check beasts' ancestry).