As you progress through the story you will unite the 3 Vanguard members (Zavala, Cayde-6, and Ikora) and they will join all the other vendors at the Farm and will be told to speak to them.
Upon speaking to Ikora you will unlock challenges which will require you to do certain tasks such as getting multi-kills or ability kills during activities such as Strikes & Patrol. These can be checked during the activity by pressing
and will appear on the right-hand side of the screen. Occasionally if there is something else being displayed on the screen there may be a delay so be patient if they don't appear straight away.
There are 3 unique challenges per activity/area and can be stacked amongst each character so this can be done within a day assuming you have at least two characters at the same stage (confirmed by OniAegis in comments below).
List of activities/locations confirmed so far for which the challenges differ:
• Nighftall Strike
• Prestige Nightfall Strike (same challenges as normal strike but stack)
• Crucible - Quickplay
• Crucible - Competitive
• Crucible - Trials of the Nine
• Prestige Raid*
*Currently unsure if there will be challenges in the prestige raid (which I'm assuming is likely).