Volatile Organic Compounds are the organic solvents used in standard paint formulations which serve as the carrier for paint pigment. When paint dries, the odor smelled is from the evaporation of VOC's used in the solvent vehicle and the tinting pigments.
Some of the more common VOC's used in paint as solvents and preservatives include formaldehyde and benzene. Pigment chemicals can include lead, cadmium and chromium. The "fresh paint" smell we often experience are VOC emissions from volatile chemicals like diethyl phthalate and dibutyl.
Oil based paint has the highest level of VOC's. Exposure to VOC's can trigger asthma attacks, create throat and eye irritation, nausea and headaches among other health problems. Long term exposure can lead to cancer and diseases of the kidney and liver.
Because of the negative health effects of VOC's, alternative interior paints are now available that contain extremely low-VOC or no-VOC formulations.
See the tutorial: Low VOC Interior Paint and Other Healthy Alternative Paints