This is not a small topic, and different group leaders may have slightly different rules. See Basic Warcraft Looting Rules for Part 1 of the Looting Rules series.
If you're not familiar with playing the game in a party or dungeon instance situation, feel free to admit you're new to the experience and to ask questions. You're better to err on the side of caution than to take action that may strongly impact your invites to future parties within your guild.
I'll try the short version, since there's so much to be said. Feel free to ask for clarification on unclear points.
General
You're in a party, which means that all resources which you can have to yourself when you play alone, now need to be shared:
Treasure Chests
Mining or Herbalism nodes
General loot from killed enemies (including locked boxes and gems)
The "/roll" function is what is used to determine who "wins" the opportunity to open Chests on the ground, or mine ores from a mining node, or pick herbs from a herb node. In this case, one member of the party who sees the chest will say "roll for chest" and all members are to /roll. The winner is the one with the highest roll, and the second-highest roller can clean up any remains the winner didn't want.
Of course, in terms of mining and herbalism nodes, if you're the only Miner/Herbalist in the group there is no need to roll, but if there is more than one Miner or Herbalist, /roll is the fairest and easiest-to-remember way to determine who wins the commodity.
Group looting of killed enemies is a little bit more complex, with the introduction of "greed" and "need" rolls.
General loot from enemy drops
Loot from enemy drops already has a system in place in terms of the rolling of "need" or "greed" upon pickup for Uncommon+ items in the standard "group loot" setting, but this system isn't perfect.
Here are the rules regarding rolling on enemy drops:
Rolling "Need" is only acceptable when:
You are rolling for your current character (no alts).
You are rolling on armor or a weapon which would be a upgrade you intend to use to what you're currently using.
You are rolling on a (new) recipe or component** used in a Profession your current character is in.
All other times, players should be rolling "greed" or passing on the roll entirely.
**In terms of components used for Professions, keep this in mind:
Treasure Boxes and Rogues or Blacksmiths
When it comes to drops that are treasure boxes (steel lockboxes, etc), ALL members should be rolling "greed" or passing. Even though only Rogues and Blacksmiths can unlock these boxes, its the contents of the boxes you're rolling for.
Gems
Lots of professions can use these - not just Jewelcrafters. Plus, gems are a good source of cash for players in general. Roll greed and do negotiations afterwards.
Enchanters and disenchanting
Rolling "need" just to disenchant objects is severely frowned upon. Enchanters should be rolling "greed" on anything they won't be directly equipping to use, and since the majority of drops won't be "needed" by someone, the Enchanter will get a fair chance at winning the loot and thus being able to disenchant it at that time, while other players will get a fair chance at winning the loot to give to alternate characters or selling in the Auction House.
Summary
The rules are in place to make it fair for all party members and give everyone a chance for not only useful equipment and the completion of quests, but also the chance to pick up some extra loot that can be sold, traded or disenchanted.
This is the end of Part 1 of the Basic Warcraft Looting Rules series. Many party leaders will expect party members to understand and operate by these rules for any PuG (pickup group) formed.