How do I make band names appear on iTunes/Music? On the right are songs I've purchased on iTunes itself, and on the left are songs I've bought (yes, I paid for them/supported the bands) from other sources, namely their websites and a code from a tour.

How do I make the bands' names appear (Undercast and Sick Puppies, respectively) below the album names as they do on the right?

Thanks!
Select the songs in iTunes, then Get Info (⌘I) and add the name of the band to the Artist field.

If you want different tracks to be by different artists, you can unify the album under a single artist using the Album Artist tag.