The BANDANA  is probably one of the most casual ties.  Its back ground in the Middle East and South East Asia around the 17th century makes is one of the key pieces from there regions incorporated into classic mens' style.  Its association with cowboy culture and gay subculture gives is a casual and anti-establishment vibe.  It is one of the few ties that does not require a proper collared shirt and can be worn with a Polo shirt.   There is some overlap with the neckerchief, which often tends to be smaller in size, although many use these terms in an overlapping way.