Any check that is a check of either of the two stats on your Path card.
Eg: for Marshal, you can burn Mythic Charges on Strength or Charisma checks (after applying the number of them as a passive bonus to your check.)
So you're making a Cha/Diplo on Alain, and you have 3 charges; you get 1d8 + 3 (skill mod) + 3 (charges) - and burning a charge, your 1d8 would become a 1d20.
You're making a massive Reveal and Topdeck Donahan Lance attack as your first combat in the turn: 1d10 + 3d8 + 2 (Melee) + 3 (charges) - and burning a charge, your 1d10 (your best die) would become a 1d20. (Burning a second charge, one of your d8 will become a d20 for 2d20 + 2d8 + 5.)
Note that I'm not factoring in Skill feats that would obviously be present as well. :)
Oh, I thought I could use it on any check as well.. So I could burn a charge as Mythic Marshal to get a 1d20 on a Knowledge check. Good to know.
It's actually pretty clear on the card. The power adds your mythic charges to a check of that type, THEN allows you to remove mythic charges. Therefore, if you were unable to activate the first part of the power, you never have a chance to use the second.