What kind of minerals have no cleavage plane?
I'm pretty sure a more rigorous answer deserves to come along, but I can give a simple overview of some of the important factors.

Cleavage planes have to do with bond strength and bond geometry.  If there isn't a plane of bonds that can be cut through, then you won't get cleavage.  When a mineral is fractured, the fracture "wants" to take the path where the total energy required to break all of the bonds along that path is at a minimum.

In Halite, you can "cleave" a perfect plane through weaker ionic bonds (the geometry works, and the bonds are weak).  In quartz, all of the bonds are of the same strength (strong silicate bonds), and the geometry of the mineral is such that you can not find a plane that will cut through all of the bonds, therefore the fractures must "wander" creating the conchoidal structures.  In mica, the cleavage planes form between the silicate sheets.  The silica bonds are strong, the the sheets are held together by weaker ionic bonds.