Binder is the workhorse leaf. After the bunch is formed — entubado, accordion, or another method — the binder is wrapped around it so the filler stays put on the rolling table and in the mold. In Spanish this leaf is often called capote.
It has to burn. A binder that will not combust makes a cigar canoe, tunnel, or go out. It also has to be strong enough to hold a bunch without adding a lot of flavor of its own, though some binders are chosen because they do add a note. A double binder is two binder leaves, used when one is not enough or when a blender wants a particular combustion or taste.
You rarely see the binder on a finished cigar. That is the point. It is construction, not costume. See wrapper and filler for the other two members of the usual three-leaf recipe.