Question

What are the lines called that run around the earth in an east-west direction?

Answer 1

The lines often drawn on a map or globe that look like east/west paths are lines of constant latitude.  The points on any one line are all at the same latitude, and there is a point on the line at every possible longitude all the way around the Earth.

Since the lines of constant latitude are all parallel to the Equator and to each other, they're often called 'parallels' of latitude.