Sections: Mid, Support, Offlane, Carry
Those on borders between sections can generally be played as other roles as well (usually the neighboring role).
Each row within each section usually has a theme as well:
Mids: Lane Dominators, Playmakers, Pushers, Melee, Mid/Support
Supports: General Purpose, Silencers/Roamers, AoE Ultimates, Terrain Control/Misc, Healers
Offlane: AoE Control, Single Target Lockdown, Teamfight Specialists, Misc., Greedy/Pushers
Carry: Manfighters/Misc., Strength, Ranged, Assassins/Escapes, Pushers/Illusions
Spots saved for Techies (next to Venomancer), Oracle (between Abbadon/Chen), and the unreleased Pit Lord (between Lich/Sand King...also because I had one extra slot).