Challenges are issued at the start of the combat round, before any blows are struck (but after Impact Hits). Only one challenge can be issued per close combat – the side whose turn it is has the opportunity to issue a challenge first. If that side chooses not to, then the other side can issue a challenge.
The player issuing the challenge nominates one of his character or champion models to issue the challenge. If he does not have a suitable model in the fight, a challenge cannot be issued. Similarly, if there are no characters or champions in the enemy units, a challenge cannot be issued – there's no one to fight!
Quite how individual warriors issue their challenges varies from race to race. An Elf or Man might salute his opponent, whilst an Orc bellows insults at his foe. To issue a challenge, choose one of your characters or champions in one of your units in the combat – this is the model that issues the challenge. Proceedings will be enhanced considerably if you actually frame a suitable challenge, perhaps along the lines of "Who's a- comin' out tae fight me, ya scurvy, no-good, cowardly rat-infested spawns o' unmentionable descent. I can smell ya fouled britches and hear ya knees a-knockin' together with fear!" Once one challenge has been made, further challenges cannot be issued in that combat.