A neck strap is any strap secured around a horse's neck. These are often used to support a martingale but are sometimes seen on their own. (In the latter case it's not uncommon for the "neck strap" to actually be a spare stirrup leather).
Neck straps on their own are used to give an extra grab handle for novice riders, usually in English. They're especially important when first learning to jump and also if the horse has a hogged or roached mane. (Yes, we do grab onto manes. It's not a problem as long as you grab a thick chunk of it).