VOCs, as BLHaas said, are "volatile organic compounds"; that is, compounds that have a high vapor pressure and so readily evaporate into the surrounding air, and are mainly hydrogen-oxygen-carbon. The ones we smell and identify as being VOCs are most often "aromatics", which have one or more six-carbon rings (phenyl/benzyl, napthyl, etc). These aren't all bad; the smell of cooking meat, of flowers, herbs, fresh-cut grass, lemons, and yes even onions, are all VOCs. The VOCs in paint are what you smell, and also what are bad for you (at least some of them), so theoretically, getting rid of the smell means you've gotten rid of the problem.
The VOCs typically associated with latex and oil paints are mineral spirits, typically an amalgam of 7- to 12-carbon hydrocarbons produced by petroleum refinement. Oil paints require a high percentage of these in order to properly dissolve the pigment and achieve the desired consistency. Latex paints are based on an acrylic resin, which for many decades now has required relatively low VOC content in favor of a simple water base. However, acrylic acid, the basis of the acrylic resins holding the pigments and binders of the paint, can easily degrade into another example of a stinky organic chemical class; aldehydes. Aldehydes are caused by hydrolysis and subsequent hydrogenation of carboxylates, of which acrylics are a member. The simplest one is formaldehyde, and it's not great news; it's carcinogenic and acutely toxic in sufficient concentrations (higher than those in the average freshly painted room). Aldehydes also have a relatively low odor threshold; humans can detect aldehydes in concentrations measured in the low parts per billion. So it doesn't take much to stink up a room.
To actually answer your question, the most common trick I could find was to cut an onion in half and place each half, cut-side-up, in opposite corners of the room. The chemistry behind that is apparently inherent in the active ingredient of onions that irritates your eyes when you cut them. This ingredient is produced by enzymatic alteration of amino acid sulfoxides, and is called Syn-propanethial-S-oxide. Notice the "thial" portion of the name. Thials are to aldehydes as thiols are to alcohols; the switch from an oxygen to a sulfur, as with thiols, typically increases a thial's reactivity and lowers its boiling point compared to an aldehyde. So, you have a highly reactive sulfur-based chemical that yearns to evaporate into the air and irritate your mucous membranes.
Another thing it will apparently do, interestingly enough, is undergo a dipolar cycloaddition reaction with aldehydes, like the ones produced by evaporation of impurities in acrylic paint. This rearranges the thial and the aldehyde into a combined substance with a five-membered, oxygenated ring, which ostensibly does not offend the senses as much as either precursor. Because aldehydes are offensive in such low concentrations, it similarly doesn't take much Syn-propanethial-S-oxide to remove them and reduce the level below the odor threshold.