Showdowns are special missions that will draw out a Warchief. Each Warchief has their own, random Showdown that must be completed every time said Warchief needs to encountered.
Assassination
"Draw out [the Warchief] by killing his Supporters from the shadows."
A number of Uruks will be marked as "Supporters' and will appear yellow in Wraith Vision. Kill several of these Uruks while undetected to draw out the Warchief. There are many, many more of them than you need to kill, and they respawn pretty quickly. The only way you can really botch this is by dying, but as long as you're careful and don't let the alarm be raised on you, this shouldn't be too difficult.
Cry Havoc
"Slaughter waves of enemies to force [the Warchief] to face you."
To draw the Warchief out here you'll need to provoke an Uruk into sounding the alarm. Just pick a fight, and certainly one of the Uruks will run off to do this. After the alarm has been raised your next task is to kill 35 Uruks. Once done, the Warchief (and his retinue) will arrive.