Historically, in cocktail bars, the size and shape of the glass determined what drink was supposed to be drunk out of it, as explains Alex Orwin, group bars manager for The Rushmore Group.
Martini glass - Designed to serve small mixed drinks such as a cosmopolitans or daiquiris. The stem ensures your hand doesn't warm the drink, so these should never be served with ice. The shallowness shows it's to be drunk quickly, so no straw either.
Hurricane glass - Bigger than any other cocktail glass. The stem is too small to hold, so the drink must be served with ice, otherwise your hand will warm it up. A tongue-in-cheek glass, drinks in it shouldn't take themselves too seriously.
Rocks/Old-Fashioned glass - Used for strong drinks served in small portions that would be stirred down with ice to open them up. The ice "softens" the alcohol burn as you drink it. Traditional cocktails can also be served in these glasses.
Collins glass - Long and slim, it's designed to hold ice and take lots of liquid. The length is significant here as it allows for more soda water or mixer, so makes for a much lighter, less alcoholic beverage.
Highball glass - Shorter than a collins glass, but taller than a tumbler, this is the most used in a bar. These glasses are long, but not so big that there will be an over-dilution of the base spirit. The mixer is supposed to be there to showcase the liquor, not swamp it.