A highball glass is quite simply traditionally used for "the highball" a drink consisting of whiskey and club soda. Highball differ from soda glasses in that a highball holds 12 oz. and a soda holds 16 oz.
Highballs are nice in that they are larger than old fashioned and therefore holds up to 4 oz. of ice, and 8 oz. of drink. They are specifically for drinks that are known as "tall drinks", most of which are poured over ice and topped off with soda, thus making the highball the perfect glass since it has room for more than two ice cubes.
A highball itself is a cylinder shaped glass with a heavy bottom. Usually soda glasses and highballs look very similar (which is the case where I work, and people are always confusing them when they try to sort them for dishwashing). The difference is, and will always be that soda glasses are bigger. So next time you get a drink, analyze the glass the bartender uses, you might know more than the wait staff does if you can pick out a highball.
Thank you Ouroboros for the added information.