You've asked a few specific questions so I will go ahead and answer those.
what is the boot drive? Is it simply whatever drive I have the Linux OS on?
/boot is the place everything your computer needs to boot is stored, including the kernel and grub. Ubuntu doesn't remove old kernels by default so they can pile up and eventually take up all the space on /boot.
I can see in my System Monitor that my boot drive is pretty much full, but the internal 1 terabyte HDD has barely been touched. Is my boot drive a different storage unit in my computer I don’t know about (perhaps something that comes default on the motherboard)?
On a basic install if boot is full your hard drive should be too, so without more information about how you setup your Ubuntu install I can't say why boot is full but your hard drive isn't. It is possible you setup /boot to be on a different partition or another HDD altogether, in which case you would see this behaviour.
I’ve done some looking around on forums and see that people clear out old kernels. I don’t quite understand what this means. Are they simply deleting the old updates, because the newer ones have essentially more efficient and better programming/capabilities?
Pretty much, yes. Periodically the kernel will be updated to a new version. When this happens Ubuntu doesn't remove the old version(s) because sometimes a newer kernel can cause stability issues with some hardware so they leave the older versions to fall back to just in case (this has only happened to me once in ~5 years).
Removing the old kernels is definitely the best solution. The best way to do this would be to periodically run the command:
sudo apt-get autoremove
which will remove old obsolete packages that are no longer needed (including old kernels). See this question for other ways to remove old kernels: How do I free up more space in /boot?
Is there a way to do this without using the command line?
Absolutely. Check out this answer, using the synaptic package manager, as well as this answer, using Ubuntu Tweak. (both from the question I mentioned in the last point)