Since you have your catalog and images on an external hard drive, all that is necessary is for you to install Lightroom on the new computer, connect the EHD and double-click on the catalog. But if you have any further questions, this article should answer them for you:
And I'm assuming that if I make changes to any images, when I move the external HDD back to my desktop, then all of those will be transferred over?
Basically, I know that LR uses non-destructive editing - just wanted to know where the "changes" and edited settings are stored. Do they reside on my desktop's internal HDD or on the external HDD?
Any work that you do on images in Lightroom is stored in the catalog. The images themselves are never touched. So if you continue to use the same catalog on both computers then all of your work will always be intact. The only thing that could create a little stress is if your monitors are not calibrated properly, and you have display differences on the two computers.