The calling-drops system was implemented by the community to better manage how loot is distributed, since the nature of the class-change system can make claiming items otherwise complicated.
Typically, a person says 1/2 items that they'd like to roll "Need" on (1 or 2 depends on the group/trial; you're more likely to see 2-call groups in the 65/70 trials than in the 50/55/60 ones from my experience), and someone makes a list of what items have been called and by who. The items that you call are expected to be the only items you would roll "Need" on, and everything else, you'd roll "Greed." Rolling "Need" on an item that you did not call can have severe reputation consequences (people are sometimes deemed "ninjas" for rolling Need on an item that another player has otherwise called; people don't like having players like those in trial groups, as the appeal of trials is typically the rare drops gained)
There can be other rules depending on the party (such as no double-calling to prevent loot competition, you're only allowed to win one item via Greed rolls, or calling a certain item in a 2-call group will count as two calls). If in doubt, ask first.
If you don't know what drops in a given place, this link
should cover a majority of the special drops. (change the "50" in the link to whatever level trial you're working on for different lists besides the L50 one)
As for actual trials themselves, the tactics on how to complete each dungeon are different depending on the trial. If it's your first time in a given trial, or you are unfamiliar with it, it wouldn't hurt to ask your party for details.