There are essentially three main issues that we try to address in creating a mouthwash: preventing dental caries (cavities), preventing gum disease (periodontitis), and improving bad breath (halitosis). Since keeping proper oral hygiene is tricky with braces, will deal with these questions often in orthodontics.
The majority of dental mouth rinses focus on either trying to address preventing cavities (via fluoride) or preventing gum disease (via various compounds). An effective mouth rinse will have either sufficient fluoride to help reduce cavities or have adequate power to kill the bacteria that cause periodontitis. Preventing periodontitis is usually the harder of the two, since killing bacteria that are sequestered in colonies in dental biofilms is quite difficult. In addition, the rinse must be tolerable for the patient, safe enough to ingest in small quantities, work over a fairly short exposure time of less than a minute (who wants to swish for 15 minutes!), and penetrate biofilm at a concentration sufficient to inactivate these hardy bacteria. Listerine is a common brand name most people are familiar with (though any generic will work the same) that can help with gum disease, but it won't help significantly in cavity prevention like a fluoride rinse. The active ingredients in original Listerine consist of four different alcohols. A common prescription mouth rinse for gum disease is called chlorhexidine gluconate. Several other compounds using other active ingredients like iodine have also been released in recent years and show promise.
Most mouthwashes assist in helping resolve bad breath to some degree (though some claim to focus specifically on improving breath). The best way to improve breath is through removal of bacteria via getting periodic dental cleanings, daily home maintenance via brushing and flossing, and really any rinse. I typically find bad-breath-only rinses to be kind of a waste of time.