Assignments are objective-based tasks featured in Battlefield 3 that reward players with weapons, camos, or dog-tags upon completion. Every assignment that doesn't require time spent on their respective DLCs can be completed on maps outside of them. Certain assignments require others to be completed before any progress can be made.
Armored Kill introduced five new assignments which were revealed on September 3rd, 2012. Since the pack contains no new weapons, the assignments unlock vehicle specializations and unlike most assignments in other expansions, these can only be completed on the respective maps.
The Professional Russian assignment most likely refers to the popular YouTube uploader, FPS Russia , who jokes that he is a professional Russian on multiple occasions, even having a shirt available to purchase with the phrase.
The Scarred Veteran assignment takes a noticeably longer time to complete for many players due to the need to play a total of four hours on two specific maps. The map-specific challenges are unique to the all-kits weapons (but only for the Back to Karkand expansion pack).
If you complete an assignment in the top row (or first per class), you get 2500 bonus points. If you complete one on the bottom row (second per class), you get 5000 bonus points.
The weapons featured in the Battlefield Premium assignments (F2000, PKP Pecheneg, L96, SCAR-H, L85A2, LSAT, SKS, MTAR-21) are the weapons that are allowed to equip camouflage. The assignments unlock a second camo pattern, with the first provided as part of the Premium package.
The filler text in the Assignment images on Battlelog is taken from Battlefield Premium marketing material, and is completely unrelated to the weapons or items pictured.