#Carry roles:
Positioning/skillshot dodging (they're almost the same things)
CS
Wave Control
Knowing when to push objectives (or in other words, objective control)
Teamfighting
Warding
Counter Picking
Positioning is VERY important because you really should not die at all, on top of having less tools to defend yourself if you get caught. The less mobile/squishier you are, positioning becomes much more important.
Because how much gold earned is important, last hitting makes rather a big difference. You can go 0/3/0 and have 100 cs while the enemy is 3/0/0 and have ~70 cs. Since ~15 cs = 1 kill, you are actually not that far behind as compared to the enemy laner.
Wave control lets you manage your gold gains, tower lives (very important botlane, leave at wrong time = RIP 3/4ths of your tower), and of course, when your other teammates can gank your lane. Also if controlled properly, you can easily deny and zone the enemy laners so YOU get even more gold.
Warding should be everyone's job, but carries should be more focused on just not die and do their job: deal damage.
#Supportive roles:
Objective control
Warding
Positioning/dodging
Teamfighting
Wave control
CS
Counter Picking
Objective control, if anything, is the most important deciding factor whether a game will win or lose. Because objectives, not kills, win you games, knowing how and when to get those are very important. You leave a lane unattended to a big fat minion wave, there goes your towers, and more openings for the enemy team to snowball. If you're behind, yes, you can learn to control objectives and turn shit around. Know how to do it, easy wins, even if your entire team is feeding and has garbage KDAs.
Warding ties a lot with objective control, because having access to where the enemy team can mean a good deal of advantage. Frost Queen's Claim counts.
And again, you don't have to know how to teamfight, but at least learn to dodge skillshots. Prevents a lot of the stupidest mistakes.
Wave control and last hitting, leave that to your laners. Though please do be aware of how they work so you don't screw up your laners.
Counter picking. You can be in the worst, hyper-counterpicked matchup, and can still win because you know objective control. Counter picking is mostly for laning, which only constitutes 1/3 of the game, and isn't even that big of a deciding factor. A counterpicked matchup can lose lane, and easily win the game because they know how to do everything else above. Even in teamfights, you still got 3-4 other teammates to back your weakness up, and even in a "completely countered" matchup, there's always objective control that matters so much more. If anything, you should rely on your skill and knowledge on how to play around a champ anyways, so counter picking is not really that big of a deal.