To veer is to make a sudden turn, like when a driver veers off the pavement or a pleasant conversation veers off in a troubling direction.
When you make any quick change of direction you veer. You can veer toward an attractive person at a party, leaving your friends mid-sentence. You veer away from an oncoming truck on the highway to avoid getting crushed. Originally, veer described a change in the direction of the wind, but as you can see, today, anyone or anything that changes direction veers.