Originally, cocktails didn't have a "dedicated" glass. Bartenders would use whatever glassware they felt was appropriate that they had on hand. It wouldn't be until the 1900's that a glass would emerge that would be seen to this day as the quintessential cocktail glass. Today we commonly call that a "Martini Glass" in honor of the drink that typically would have found it as its home.
It is difficult to find exactly when/where this glass first appeared, but there are a few rumors that point to the 1925 Paris Exposition as being the unveiling of the glass which would soon become a Martini Glass. So far, we have found very little about this, but after some searching we located an image of a set of glassware by René Lalique which had been displayed there.