Reset Timer - unlike the other replayable scenarios, this timer starts counting when you begin the Challenge. So if you spend 20 minutes playing you only have to wait 2 hrs 40 min, instead of the full 3 hrs.
Attempt - an 'Attempt' begins when you start a Challenge, and ends when you Exit that Challenge (or die), this has special significance, not only for the Reset Timer, but also for the Wand Of Dragonfire
Level - in this context it represents 1 plus the number of consecutive victories in the challenge.
Victory - defeating, fleeing, surviving, or exiting a Level. Your last Victory on an Attempt determines the highest Level achieved for that Attempt.
Starting Level - the level you begin at after successive Attempts. Starting Level seems to be decided based on the number of Attempts you've made on that Challenge. For more, see Talk. You have the option to start at this 'Starting Level' or at level 1. Choosing this option will not reset the 'Starting Level'
Increasing Difficulty - each Challenge increases in difficulty after each victory. The type and size of the increase varies between Challenges.
In this context, a Random Encounter represents the special encounters you can have. At the beginning of each Level (including the 1st) you have a chance to have a Random Encounter. The Orbs and the Ghostly Young Woman are available in both Challenges and in the exact same way (i.e. the difficulty and available rewards of each is the same across both Challenges):
Orb of Destruction - A Black Orb
To succeed at touching a black orb, you must beat 75 on 1-100 + Spirit + Aura. Best success chance = 65%.
If you pass the check you get 16 general xp and "feel somewhat strange." For a certain amount of subsequent combats (2-8 attested) the following message is displayed upon your first attack: "Your vision blurs as a powerful pulse of negative Nevernal energy leaves your body...and decimates your foe..." and your enemy is killed instantly. This is followed by a message telling you how many pulses you have remaining. This ability does not use your Nevernal Reserve.
The black orb does NOT work outside this challenge so it cannot be used to finish that tough quest you have been trying to complete.
Sphere of Healing - A Blue Orb
To succeed at touching a blue orb, you must beat 75 on 1-100 + Agility + Body + (Luck * 0.5). Best success chance = 75%.
If you pass the check you get 16 general experience and your SP is restored to full
Level Skip - a portal that allows you to bypass a random number of levels (2-6 attested) as a reward for a certain type of victory. The type of victory you must achieve varies across the Challenges. Also, the behaviors of the Level Skip are different between the Challenges.
Other - some Challenges have unique random encounters, such as the Ghostly Woman in the Bone Horde Challenge.