Broadly speaking, peer seems to have two meanings, looking intently and being partially visible.
a). She peered into the darkness.
b). The moon peered from behind dark clouds.
However, I have difficulty understanding ‘peer’ with ‘around something’ as follows. Would you help me? (‘Around’ often annoys me!)
1.
Go, Harry pleaded with him silently, go with Snape … go … Mrs. Norris was peering around Filch’s legs…. Harry had the distinct impression that she could smell him. (Harry Potter 4 [US Version]: p.470)[Bold font is mine]
N.B.: Mrs. Norris is a cat. Harry is in an Invisibility Cloak.
2.
All of them were looking at something Harry couldn’t see and sniggering heartily. Pansy’s pug-like face peered excitedly around Goyle’s broad back as Harry, Ron, and Hermione approached.(Harry Potter 4 [US Version]: p.511)[Bold font is mine]
PS: *No.1:* I thought at first the cat is just walking between somebody’s legs like forming a figure 8, but I couldn’t find a good example on the Net or YouTube. Or is the cat looking at a specific thing, for example, Harry as a smelling point or Filch’s legs? Or, is it similar to ‘looking around something’: to visit somewhere, walking around it to see what is there? (Unlikely… well, the sentence pattern seems similar, though…)
No.2: I wish I could find any dictionary says ‘around’ means ‘from behind’.
Best Answer
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...