There are three generalized carry styles based on who you're playing.
DPS Carries (ex. ADCs) generally want to always be in position to take or defend an objective. DPS Carries are the best designed for killing/destroying big things, and as it turns out, most turrets, dragons, and barons can all be considered big things. This can be considered in many forms. If you see the rest of your team grouping up around a tower or one of the epic monsters and you're a ranged ADC, you almost always want to be there to help them. Most Melee Carries want to split push because they are so fantastic at dueling, and with an Elixir of Ruin and a bit of attack speed, they tend to shove lanes and threaten towers at phenomenal speeds. The last example is a Jungle DPS (Master Yi for example) may want to solo the Dragon while nobody is looking.
Burst Carries want to do their absolute best to shut down enemy Burst and DPS Carries. If you're playing Zed and the enemy team's Caitlyn is fed, you need to make sure that Caitlyn is never alive long enough to help in any team fights or objectives. I highly recommend being a major helper to Vision control, as this can help you individually catch anyone on the enemy team who happens to be running around on their own and will help your team as a whole. You may also want to shot call, since your attention is already on searching for enemy carries, and you know which objectives they can or can't reach.
Tank Carries want to defend their DPS (usually) and establish zoning control. They want to invest in vision to make sure enemy Burst Champions can not flank their team and threaten their DPS. They also provide the straight up presence, usually in the form of CC, to give you advantages in objective fights, for example handling turret aggro if your team decides to dive. Their job is to assist the DPS in taking objectives, and carrying as a tank is all about making sure your DPS lives longer then the enemy team's DPS.
There are more specific tips for specific champs, and there are more generalized tips for all champs, but these are the three general categories.
