face-on galaxy and edge-on galaxy
I already googled a face-on galaxy and an edge-on galaxy, but I couldn't find any proper information. Would you explain them to me?
An edge on view of a book is simple, a face on view should be logical. spiral galaxies are flat.
The terms refer to the viewing angle, i.e. from which direction do we observe a (disk) galaxy. If we happen to be located roughly in its plane of rotation, we see it from the "edge", whereas if we are more or less above or below its plane, we see its "face".
To make a more quantitative statement, the orientation is described by the inclination angle $i$, which is the angle away from "face-on". That is, the inclination of a galaxy that is observed face-on is $i=0$, and the inclination of a galaxy observed edge-on is $i = 90º$. For randomly oriented galaxies, the mean inclination angle will be around $i = 60º$.
Wrt. your second question, you're right that the text is not correct. The three small constellations lie north of hydra. "North" means "toward higher values of declination". Perhaps it's more easily seen in this image (from Wikipedia):
Regarding the north/south issue, I think you're correct, but you wouldn't believe the crazy inconsistency between the meaning of north/south/east/west in astronomical coordinate systems that I've seen. Maybe the book is "right" using whatever weird way of defining south that they have.
@zephyr: I agree, astronomical coordinates are a mess to get one's head around. But in general, if you see a map — at least in scientific journals — north is up, and east is left (i.e. R.A increases to the left). This is contrary to a world map where east is right, since the sky is seen "from the inside".
I just want to add pictures to augment pela's answer.
Just google "edge on galaxy" and "face on galaxy" to find results like these, it is entirely about the viewing angle.