Approachability is probably the most neglected thing in business headshots, but is very important. Business headshots must show a good mix of confidence and approachability for them to be great shots. If you can bring these two emotions out, you'll have a winning shot.
For actors, it's a bit of a different story because you are going for different looks, and you want to bring all different kinds of emotions out. The idea of approachability is a bit more of a character look, rather than the overarching two themes of confidence and happiness. Approachability means the following:
As you can see, the whole idea of approachability is basically about being likable. You want someone looking at the shot to think of you as the person they can come to to ask a question to, or the person they don't need to fear asking for help. You want to be a nice person, someone who isn't hiding anything or isn't mean. Likable people typically have softer expressions and relaxed faces.
The reason you might want an "Approachability" shot is because you may want to play a friendly neighbor, or someone to cry on. If you submit the wrong headshot to the role, it won't look like you can play that role, because approachability is much different than confidence, happiness, or the other emotions.