There are 10 stages total (yep, only 10!). The goal of each is to touch the "yellow thing that may or may not look like a crystal ball, but let's just say that's what it is" (that's literally what the manual says!). Each step you take leaves the floor tile you were just standing on darkened, however, and you can't cross back over darkened ground pieces, so you'll have to plan your route carefully. And in some stages, there are monsters that will kill you if they touch you, so you'll have to study their movements and work around them as well.
Although you don't strictly have to cover every floor tile to beat the stage (simply touching the crystal ball will do), each stage ends by displaying a unique piece of art featuring the game's heroine, and any floor tiles left undarkened will block out the corresponding portions of the artwork. Additionally, the artwork for that stage will not be added to your "album" (viewable from the title screen) unless every floor tile in the stage has been darkened before touching the crystal ball.
If you get stuck or just plain decide you want to redo the current stage, you can hit Esc to restart it.
You have magic at your disposal to make things easier, which is simultaneously the coolest part of the game and the most underutilized. Magic spells each require multiple specific crystals to cast, which you can select by pressing Space. Move the cursor over to whichever crystal you want to add to the mix and press Space again to add it; as long as you have at least 1 of that crystal available, you'll see the number under the crystal decrease, indicating that it's been successfully added.
Once you're done selecting the crystals for the spell you wish to cast, press Enter to cast it and -- as long as you selected a valid mix of crystals -- choose which direction to cast it in, or what floor tile to cast it on, pressing Space again to finalize the spell.
If you selected a mix of crystals that is NOT a valid spell "recipe," nothing will happen, and -- worse still -- you won't get back any of the crystals you used, meaning you'll have lost them attempting to cast a failed spell. You gain more crystals at the end of each stage, however, so you'll be able to try again soon enough.
You'll also learn recipes for new spells at the end of certain stages, though you CAN cast spells before you've learned them, so feel free to just experiment -- you'll lose lots of crystals in the process, but you might discover a useful spell or two just by fooling around.
If you can't read Japanese, don't worry, as the text that teaches you new spells is always the same, indicating that you found a scroll with a new spell listed on it. The second line of this text box indicates the spell name (which is always written in English, making it easy to pick out from the surrounding text), followed by the exact mix of crystals required to cast it, so just look for the tiny crystal symbols and, a couple characters after each, a number -- and there's your recipe!
As an example to fool around with, the first spell recipe you learn (revealed at the end of stage 2) is Fire Ball, which requires 3 red crystals and 1 green crystal (see the annotated screenshot included in this zip).
NOTE: There's at least one spell combination that crashes the game, that being 3 of the third crystal. I guess that's the special Error spell! Nobody likes that one. ;)
The game may also crash if you sit on the victory screen for a stage (the screen where the spell recipe is shown) for too long.
Source: Wyrdwad in Japan (Twitter)