Curing and drying are two different processes in modern paints.
When paint dries, the carrier (typically water in latex paints, a solvent in oil based paints) evaporates, allowing the paint to form a film. As the carrier evaporates, other chemicals that act as stabilizers evaporate as well.
The stabilizers prevent the paint curing in the can. Once they start to evaporate, other chemicals in the paint are free to react with each other to form new chemicals. Usually these are small chunks of polymers (think plastic) combining to make larger chunks of polymers that effectively form a large sheet.
The paint will not achieve its full strength until it is completely cured. It will be easier to scratch or stain. Under good conditions, paint will usually be 80 - 90% cured in a couple of days, fully cured in a few weeks.
Also, as the chemical structure of the paint changes, the color and opacity of the paint film will change as well. If you put new paint over old paint of the exact same color, the colors will look different for a while after it is dry, but as the paint cures the colors will come back into line.