The badges are descended from Macharius's ancient campaigns, but they lost their meaning after 2nd edition and are now just generic symbols.
(Red) Upheld fist: 5th army group under General Crassus
White Wing (on blue square): 7th army group under General Cyrus
White skull on black diamond (originally an oval): 2nd army group under General Sejanus
Note that the geometic symbols (red diamond in black square, yellow square in blue square, yellow triangle in red square) were meant for use with artillery vehicles.
I think the stars was a parody take-off of the American star, and the sickle a parody of the Russian hammer & sickle. Not sure if the axe (with lightning), the lightning bolt or the fist holding a lightning bolt are meant for, but I'm betting they were parodies of WW2 country emblems (I'm guessing the lightning bolt may be a covert "SS
" for Germany, and either the axe or the fist being Great Britain).
<EDIT> Glancing through the 2E Imperial Codex, I found this:
ARMY BADGE: Every piece of equipment and, in theory, every trooper's uniform, should display an army badge. This insiginia assists in the administration and movement of vast amounts of military hardware and personnel deployed through the galaxy to help prevent "friendly fire" incidents. (pictures reference the mailed fist, skull in black diamond, lightning bolt in white circle, fist with red lightning bolt, and axe with lightning as Army Badges).
COMPANY AND SQUADRON MARKINGS: Each Imperial Guard tank company is distinguished by two main features: a series of broad stripes painted onto the hull or turret of each vehicle in bright contrasting colours and a symbol. Each squadron in the company displays a different coloured version of this symbol. (pictures reference the star, sickle and wing as company/squadron marks for tanks, the "C", "1", "2", & "3" in the bisected circles for Chimera-hulled vehicles, and the geometric shapes for artillery pieces).
This isn't an absolute (personally, I looked into how Germany marked their vehicles for my own guard), but if you look at GW
's models, this was how they did their studio models.