Paul and Jamis in the 2000 Dune TV miniseries
A Tahaddi Challenge was a Fremen challenge to mortal combat, usually over some primal issue, such as the desire for another man's wife, or if honour was insulted, or if leadership was challenged.
Paul Atreides killed Jamis in mortal combat after Jamis invoked the Amtal Rule, and issued the Tahaddi Challenge to him.
Succession
In Fremen society, someone wishing to become the Naib of the tribe must "call out" the current Naib; issue the Tahaddi Challenge and fight to the death.
When Paul Atreides took the mantle of Mahdi, he changed this custom, refusing to call Stilgar out, as he did not want to weaken the Fremen before their battle against the Harkonnen. He also did not want to kill his friend. He made it extremely clear to the Fremen that no one man could stand against him in single combat, and that it was foolish for the Fremen to kill their best and wisest leaders for the sake of ritual.
It is unknown if this changed the pattern of Fremen succession permanently, or if they reverted to the old custom after Muad'dib became the Emperor. Either way, it became a moot point after the ascendance of Leto II - he controlled the Fremen completely, eventually forcing them to degenerate into the Museum Fremen.