Actual Gesture: Holding both hands together and lightly bowing your head
It's Function?: It's as the name says, specifically, for all the things that had to happen just for the meal to appear on the plate (which, when you think about the farming, packing, cooking and so on, is quite a lot); for a frame of reference, think about the folks who say grace before a meal(and like saying grace, this is mostly done in the home, and among company)
Any Associated Phrase?: Yes, and this one has two; 「いただきます(Thank you for the meal)」for before the meal begins and 「
Anything else?: The bow and each phrase are done at the appropriate points they ask for in order to show your gratefulness for the grub, whether each is done at all is up to what the context and the mood call for.
Literal Meaning: I didn't finish the task at hand
Intent: I'm sorry
Usage: This actually has 2 uses: as both as apology for something and a humble way to thank someone for doing something, like telling you co-worker, "Thanks for picking the slack" when you forget to do the dishes before you jet, and they cover your six by doing it before they head home for the day.
Notes: This comes from the verb to express you did finish doing something, 済む, and all the stuff you just learned also applies when it's used in it's more casual negative form, 済まない(this also includes variations of this, naturally). To specifically say you're sorry for something, you'd use ごめんなさい(the others, you'll pick up as you go)
Example: (saying sorry) パーティーに
(Sorry about getting too blitzed at your party, man. It's been a while, so I forgot I'm a lightweight with the booze)
(saying thank you)
(Thank you for the help back there. I really appreciate it)