Everyone is familiar with the Mimosa (Champagne and orange juice), and the Bellini (peach puree and Champagne), and the Champagne cocktail (Champagne, sugar and bitters) even makes an appearance from time to time. (I just watched the movie Match Point over the weekend, and one of the main characters ordered one in a bar. Though the movie does take place in London.)
My favorite "Champagne cocktail," though, is a Kir Royale. Made with 5 parts Champagne and 1 part creme de cassis (blackcurrant liqueur), it's so popular in France as an aperitif (where I enjoyed it for the first time and loved it right from the start). It's bubbly kicked up a notch--with a bit of sweet dark fruit flavor and a pretty pink color. (Made instead with Aligote, a white wine from Burgundy made with the Melon de Bourgogne grape, it's known simply as a "Kir", or as a "Kir Normale.")
**Note: When making any cocktails with Champagne, feel free to substitute other sparkling wines instead. I personally don't like masking the flavor of an expensive bottle of the real stuff with OJ or peach puree, and a full-bodied, yeasty Champagne can often taste a little odd with those flavors anyway... Spanish Cava and Italian Prosecco are refreshing and less expensive alternatives that still give you that desirable fizziness, as well as lots of clean, crisp fruit.