If you're doing quests that require you to each collect "X" number of items, report to the party your status as you go. A good way to do it is to say "4/20". That means you have 4 out of 20 items. That lets people know how far you are and how far you have to go. Also, after a party member has completed their collection, it becomes easier for the rest of the party to collect those items; any items dropped for a person with the complete set will now be available to those in the party that still need them.

If you are doing a multiple part quest and some players are behind, you could help them catch up by doing the earlier steps of the quest. That will allow everyone to be on the same step.

Buffs
Ask for a caster to "Buff" you again when the original wears off. If you're the person casting the Buffs, ask your party members to let you know when they need/want them again.

When Buffing, be sure to Buff yourself first and then the other members of your Party. When your Buffs begin to drop, you will automatically know that your Party members' Buffs are about to drop soon as well.

Party Roles
Each class often has particular strengths in a Party; see Party Roles for more information. Assign each character roles which make best use of their strengths in the situations you are likely to encounter

Don't Run!
It is important not to run, even if you're about to die, unless the whole party is running. If you run you will likely wake up more monsters or travel to some place where it will be more difficult to resurrect you. Ask for help and stand still so that the tanks can come and taunt them off you or kill the monster. Don't run in circles either because the melee players won't be able to reach the monster.

Stop attacking the monster so that you don't rack up points in their hate memory/list. You might even want to just die instead of getting the rest of the party killed. You can be resurrected once the battle is over.

Communication

Let your party know if you see a monster patrolling near that might attack unexpectedly.

Let your party know if you have to leave your computer.

Let your party know if you are low on Mana if you are required to have Mana for a battle.

Let your party know if you spot anything they are looking for, such as specific monsters, landmarks, or even trade skill nodes (Mining/Herbalism).

Let your party know if you're about to die and you need healing (if you're unsure if they will heal).

If you are a caster, let your party know if monsters are attacking you so they can stop attackers if someone isn't already drawing them off of you.

Be reasonable in your communication. You don't need to spam or repeat yourself, or otherwise annoy the players you're asking to help you.
Pay Attention/Keep Up
Don't spend time watching TV, reading, or diverting your attention away to some other distraction. The game usually requires your full attention.
Don't Wake Up Slept/Disabled Monsters
Players put monsters out of the battle to make the battle easier. Don't wake them up. Make sure you say sorry if you do and try to correct this in the future. Casters should also make sure they don't try to sleep monsters that are already being attacked.

Pool Resources
If you pool your money together you can often buy something good for a party member that will help the party. This should mainly be done only if you are playing with good friends or guild mates.