Brief (and opinionated in some parts) rundown of how to make a more concise and successful PF.

––-

If a party member joining can fit in multiple available positions, they will fill the LEFTMOST one first:

1: Any Tank
2: Any Healer
3: Any DPS
4: Any DPS

A DPS joining would fill slot 3. However:

1: Any Tank
2: Any Healer
3: Any DPS
4: BLM

A BLM joining this party will occupy slot 3 first. If that's not what you wanted you should have the most restricted slots first. 8-man example:

1: PLD
2: Any Tank
3: WHM
4: Any healer
5: BRD
6: BLM or SMN
7: MNK or DRG
8: Any DPS

Also note that party members who were in the party before the PF started are treated as having their slot be (their current job) only. If they leave, it will default to that job.

––-

Need>Greed parties will sometimes have issues filling slots where a required or helpful job has nothing of value from the instance. Examples include Bards in Turn 1 or tanks in Garuda EX. For these cases you may want to consider a Greed party; Everyone picks 1 item to greed on; anyone who hasn't declared that item passes it, and any undeclared items are usually up for grabs for anyone. There are some variations on this: some parties pick 2 items, or declare that no 2 people can declare the same item.

You will occasionally find some jerk who will Need anyways, but just boot them and move on. These parties tend to fill faster in my experience.