How do I make a Views view that groups lists into sub categories

How do I create a View which groups lists by Taxonomy Terms into separate lists on the same page.

I have created a list in Views that lists all Biblio nodes in my taxonomy vocabulary "Library". Within Library, I have several Terms. I want the list to display the list of nodes of each Term separately, with the appropriate Term name at the top of list.

So the View would list all Library nodes as follows:

Library

Neurophysiology
-Node 1
-Node 2
-Node 6

Anatomy
-Node 3
-Node 4
-Node 8
-Node 9

Surgery
-Node 5
-Node 7

All I've been able to do is list all the nodes from all terms in one big list with the node title field and the taxonomy term name listed for each one. This looks ridiculous:
-Node 1
Neurophysiology
-Node 2
Neurophysiology
-Node 3
Anatomy
-Node 4
Anatomy
-Node 5
Surgery
etc....

I'd like to do this in one View, as opposed to creating one view per Term and then knitting them together with Panels. Is it possible to do this in one view?

As a side note, I'd like to be able to choose to show the teaser field along with the title or hide it by clicking a link or button. I'd also like to create a View that just lists category names only (i.e. Term names for the "Library" vocabulary) as links to pull up a new page that lists all nodes just from that one selected Term. I realize these are questions of different, unrelated technical aspects than my main question above, so please feel free to ignore this side note if they are distracting.