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 their character models to issue the challenge. If they do not have a suitable model in the fight, a challenge cannot be issued. Similarly, if there are no characters 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 their opponent, whilst an Orc bellows insults at their foe. To issue a challenge, choose one of your characters in one of your units in the combat – this is the model that issues the challenge. Once one challenge has been made, further challenges cannot be issued in that combat.