> What are the “little square hats” called and what do they stand for?
Tefillin. Usually not seen in Reform services.
> What is the practice of bowing or rocking called and what does it signify?
Davening. (“Daven” is Yiddish for “pray”.) It’s intended to help one concentrate on their prayers. From what I’ve seen, the more Orthodox the congregation, the more wild the rocking.
Davening just means praying, and you can daven while staying stock still. The term for the swaying is shukling, although it’s not actually used much in an Orthodox context, where people tend to do it the most. Actual bowing happens a few times during the services (there was a thread on this some months back), mostly during the Amida, not constantly, and is something different.
In the interests of completeness, I will note the one time kneeling and prostration happen: On Rosh Hashanah and Yom Kippur, the additional service for the holiday includes the Grand Aleinu (the original location of the text now used as the closing prayer of each service). This prayer includes the line: “We bow and prostrate {I’m having trouble translating the third verb modi’im} before the King of Kings.” At this line, the hazzan or other leader of prayer (and in many Orthodox places, much of the congregation) do exactly that: Go down on their knees and lay out on the floor. (The same ritual is also done in some places during the high confessionals in Yom Kippur’s retelling of the Temple service, as a memorial to the people doing precisely that back in the Temple’s day.)
I was refering to the rocking back and forth and not actual bowing, but I didn’t describe it well in my post.