What does ‘peer around something’ mean?

The cat above is peering/looking/peeking around a corner or a bookcase, or whatever this non-transparent object is. She would like the rays of light from her target to reach her eyes along a curved line, going around the corner; but that isn't possible, so she has to move her head partly "around" the corner, or bend her body around the corner, to see what's going on. The cat in your story is doing the same around someone's legs. As Aedia says above, it is the same as peering from behind something, peering while you're mostly standing behind something.

Looking around an open space, like looking around a room, is different: then your head or body is rotating around an axis, so that you can look in several directions successively. There is no blocking object.

P.S. To everyone: I apologize for the cheap cute-cat trick, but a picture was really the easiest way to show what's going on, and this one was on the first page in Google Images, and, well, it is a cute cat.

P.P.S. This one is even more impressive:

P.P.P.S. Oh, I really couldn't resist this one, I'm sorry...

To peer around something means to look from behind it. The confusion may come in that your second use of peering is not actually an alternative meaning:

b). The moon peered from behind dark clouds.

This is instead making use of a literary device called personification; it projects person-like qualities on the moon, as if it were looking out from behind the clouds.