Here's a quick breakdown:
Conquest Levels - "Traditional" Marble Blast style levels. You pick a difficulty, you pick a level, you play - goal is to collect the gems and hit the finish in the shortest amount of time to get the best score.
Exploration Levels - Explore new worlds, more of which unlock and become available as your Player Level increases. Exploration levels work sort of like Gem Hunt, except the gems are randomly placed each time you load a mission (no static spawns) and the mission itself is randomly generated. The goal is to obtain the highest score, which requires a bit of skill, luck, and determination. You are not bound by a time limit, merely you collect as many gems as you wish and "finish" by rolling off the level into the next "zone", prompting the end of the mission. Your score is based on how many gems you collected and how quickly you collected them, and how large the level is.
Adventure Levels - Put your skills to the test, play a series of levels with no restarts, no out-of-bounds, and not do-overs to see how far and how fast you can progress - These levels reset on a weekly basis, and you may play each set of Adventure missions once per week. Players receive rewards for how far they progress though each Adventure mission and a bonus reward at the end of the week based on how they compare to other players' scores.
~Aayrl