When you blackball someone, you ban him from a group. A bunch of kids who kick a tattletale out of their secret club essentially blackball him.
You can use blackball as a noun or a verb — a blackball is an act of excluding or ostracizing someone. Country club members might initiate a blackball against an eccentric member who wears tie dyed t-shirts on the golf course, for example. The origin of the word goes back to the 1700s, when club members would cast secret votes on membership using white and black wooden or ivory balls. The black balls represented "no" votes.