There is a thing I have noticed in a bunch of songs where the bass guitar plays slowly and the notes go up then down and back up again.
This is one example I found where it's pretty clear but I have heard it a lot.
Is there a name for this style?
We call this the walking bass.
Although it's most common in jazz, we actually see precursors to it in the Baroque era.
In other genres, the bass often just holds long notes that belong to the chord. In a walking bass line, the bass voice moves by scales and arpeggios to add interest to the bass. It also provides some forward momentum to the music; this, combined with the steady rhythm of the bass line, gave it the term "walking" bass.