Story Mode: You'll have to help Screeny rebuild Voxtale's world and find out what happened with the Event. In order to do this, you'll have to complete all the Stages in the different islands you go through. The Stages can be unlocked with different amounts of Cubes.
Random Mode: The Hypercube is a remainder of the disaster that keeps on absorbing and disassembling everything in its wake. From it, you can retrieve random Models that you will have to assemble within a predetermined time limit. If you don't make it, the Model will be lost and may not be seen ever again. Where he takes the Hypercube, nobody knows.
If the Model is completed in due time, you'll be able to either sell it for Cubes, place it in a free slot within the Base, or put it away in the backpack.