Specifically a party-only ability, Challenging Shout is used in only one situation: A party member has generated a large amount of aggro and they need to be pulled before they die. It can also be used if a single mob has resisted [Taunt] and you need to pull that mob off your party member.
This is especially useful in instances, but even with the new 3 minute cooldown, it should not be used as a staple of holding aggro. It is best used if a given fight looks to be getting out of control, where multiple enemies are starting to split off and attack different group members, or unexpected adds join in and start attacking the squishies. Ideally, skillful use of the much faster cooldown single [Taunt] and rapid target switching should be enough to keep everyone focused on the main tank in normal fights.
As with Taunt, this ability has little effect on enemies already attacking the warrior beyond preventing them from switching targets for the duration, but unlike Taunt does not actually create fresh threat itself. Enemies will switch back to their original targets after 6 seconds, unless fresh threat is generated by other means.
With its three minute cooldown, Challenging Shout can be used roughly every third pull in an instance (assuming a minute for each pull). Because of this, warriors don't need to worry so much about the cooldown, but it should still not be used willy-nilly.
It has no real use while soloing, and like [Taunt], has no effect on enemy players in PvP.
Like Taunt it will affect player pets, allowing for a limited usability in PvP, such as pulling pets off of a caster.
Unlike Taunt, Challenging Shout has no stance requirement, so DPS warriors can use it without changing stance, in PvP or in emergencies.