Question inspired by the orbital tracking of the Chinese sky-lab: why do satellites go north and then south and then north as they orbit? Why not just a straight path?
They do go in a straight path (or at least, as close to straight as they can get). That’s why they go north and south.
Grab a globe, and hold a hula hoop around it. The hula hoop is the path of the satellite. Unless you hold the hoop lined up exactly with the Equator (which is possible, but from most launch sites impractical), some parts of it will be north, and some will be south.
Just to add to that:
And the Earth rotates under the satellite. So, when it completes an orbit, the earth has moved, which is why each pass is about 22 degrees to the west.
For a geosynchronous orbit, the satellite doesn’t have a land track that goes around the earth, instead taking the same amount of time to orbit as the earth does to rotate. If you look at the ground track of such a satellite, it makes a figure 8,
If you have Google Earth, you can see this in action. Go to some spot in either the Northern or Southern Hemisphere. Point yourself straight east. Then hit the up arrow key.
Watch what happens are you orbit the Earth. 
To put it a different way, satellites orbit in (more or less) a circle around the centre of the earth. On a map, lines of latitude (other than the equator) are not circles around the centre of the earth, so a satellite can’t follow one of those.
The moon does this as well. The inclination of its orbit to the Earth’s equator is around 18°, so it moves north and south by quite a lot over a month.
Some earlier discussion that touches on this topic: https://boards.straightdope.com/sdmb/showthread.php?t=836132
Also, by inclining the orbit and putting the satellite in a highly elliptical orbit you can produce an epicycloid ground track that causes the satellite to dwell over a narrow region for a relatively long portion of its orbit, which is useful for telecommunications or surveillance satellites intended to view high latitudes locations. So it isn’t just a limit of what azimuth the satellite can be launched into from a particular launch site but may also be deireable from a mission capability standpoint.
Stranger
That is one specific type of highly elliptical ‘dwell’ orbit but there are a number of other highly inclined orbits used by broadcast satellite and surveillance platforms like SBIRS HEO.
Stranger
The answer to the question that you are asking is that you can’t accurately display a 3 dimensional object (the x- y- and z-coordinances of the satellite) in a 2 dimensional format… The appearance that the satellite is moving in a sine wave is an artifact of the projection.
Damn, NASA took J-Track 3D down. That would have been helpful since, as I recall, it could draw orbit tracks for you.
Agreed - I didn’t mean to imply that was the only one; it was just one that I remembered hearing about when I studied orbit design.
Thanks Andy L; I thought of that thread when I read the OP (since I was the one having problems visualizing the ISS orbit in 3D). The posts by Dr. Strangelove and LSLGuy really helped me see it as it was actually happening. The graphics as presented here make it look like the ISS (in this example) is changing direction when it gets so far north or south. It is actually continuing on its circular orbit around the Earth but the only way to represent it in two dimensions is to show it turning south or north.
As the Earth rotates beneath the ISS the path shifts a little each time, which is why we see multiple paths illustrated. The ISS is traveling in what is very close to a circular orbit. The orbit is not aligned with the North or South Pole or the Equator so it looks like the ISS is changing direction every so often, but it’s not. As explained above in this thread and the one linked by Andy L, visualizing the orbit around a globe helps (at least it helped me).
Thank you! That explained it perfectly.