SJC - Mineta San Jose International Airport
This is the preferred airport. It is the closest airport and is approximately 25 minutes from Palo Alto and 20 minutes from Mountain View (no traffic). It's a smaller airport (relative to SFO anyway) and therefore generally more low-key and less stressful.
This is approximately 35 minutes from Palo Alto and 40 minutes from Mountain View (no traffic). This is the largest airport in the area so generally has good flight pricing and availability. Traffic can be pretty bad during rush hour on this stretch of 101 so if you are driving, it is advisable to take 280 during morning or evening rush hour (280 is more scenic anyway and is not much longer).
Both airports are loosely connected to Palo Alto and Mountain View by the local mass transit system, Caltrain. The SJC connection is much easier to deal with (free shuttle from airport to train station). The Caltrain schedule leaves much to be desired so it is best to rent cars and coordinate rides.