Each level in Chiral is introduced by the requirements you need to polish off before you can go on to the next level. This consists of forming a certain number of molecules that are composed of a certain number of atoms. Complete these requirements and you are on your way to the next level!
Each atom has a dot pattern representing the number of bonds it needs to form. In order to clear assembled molecules from the screen and score points, every atom in each new molecule must be "satisfied." To be satisfied each atom must have all of its bonds fulfilled. An atom with three dots must have bonds to three other atoms to be truly satisfied, an atom with one bond may only bond with one other atom, and so on.
Atoms bond automatically to other atoms that are on the playfield, you simply need to strategically place the atoms so that they satisfy each other's bonds. A molecule is formed when all of the atoms in that molecule are bonded together and satisfied. When you form a molecule, it is removed from the playfield, and points are added to your score. Additionally if the molecule was big enough (ie if it had enough atoms in it), you'll be that much closer to reaching your goal for that level.
Atoms drop into the vial periodically (how quickly they fall is dependent on the skill level you have chosen), your task is to place these atoms on the playfield so that they bond together to form molecules. The bottom-most atom the in vial is always the atom that will get put on the playfield when you click the mouse button to place it. Watch that vial though -- if the vial overflows with atoms, the game is over!