Batting Average is a Newgrounds term expressing the average score of the top three submissions of an author. As it says in the Newgrounds official Wiki:
"Batting Average is [an] average score for an author's Flash submissions. Only users who have submitted at least three Flash movies/games have this statistic. Posts, Experience, Blams/Saves, or popularity do not factor into a user's batting average.
There is a list of top batting averages which takes into account all of an author's submissions. In addition, every author has a separate batting average which takes into account only his/her top three submissions; that average is listed next to the results of an author search."
Like the Alpha section, they were affected by the '07 redesign; they became bugged and have never been fixed. Since BA is determined by your three highest scoring flashes, anyone who had three or more submissions before the redesign is stuck with their pre-redesign BA, and everyone else does not have one until the bug is fixed.
Before the '07 redesign, the BA used to be on authors Userpage and was often quoted and considered a key ranking on Newgrounds. As Highwatermark says in An Expert's Guide to NG from 2007 "An A+ Batting Average can categorize you as an expert flash author and getting this is a great achievement". However since it is not fully functional it is largely ignored and appears only on the author search, where it goes mostly unnoticed.
According to Auz it was originally an average of all of a user's submissions.