With the revival in popularity of the martini, the name itself has become a sort of "catch all." Just as all batsmen are not "Sachin Tendulkar," and everyone with a tennis racket is not "Andre Aggasi," not everything that comes in a martini glass is a martini.
According to many pundits, the only true "martini" is made with gin, an optional wash of Dry Vermouth and perhaps, a crisp, green olive (for the more daring sorts). Vodka was not introduced as an option until many years later, and no doubt gained in popularity as a drink of choice by the ever so popular James Bond. How can anyone forget “ Mr. Bond” ordering… “just a martini, shaken not stirred” Some consider that martini made from vodka is called "vodka martini." A Cosmopolitan, Cosmo, or even a Cosmo martini is amongst the feminine versions of this beautiful drink.
A "cocktail" is basically a mixture of some sort of spirits and fruit juice. Pour your cocktail into a chilled martini glass, and it's commonly identified as a type of Cosmo or specialty martini.