Traditionally, the Bellini is made with a Prosecco sparkling wine, augmented with peach purée. Traditionally, the peach purée is made of fresh white peaches that have been puréed and then strained. You then place two parts of the purée in a champagne glass (figure 1/3 of a cup at most) and add two parts of chilled sparkling Prosecco (say two thirds of a cup, or to the top of the glass). You serve the Bellini in a Champagne flute.
Some sources suggest that in Cipriani's original drink a crush raspberry or strawberry was added to the puré. If at all possible, you want to use fresh ripe peach purée. If you live in California, you can fairly easily find white peaches, but doughnut peaches will work, or any ripe flavorful peach. You don't want to use commercial frozen peaches, but peach nectar is a reasonable substitute for fresh, ripe, just puréed peaches. Other variants use peach nectar and peach schnapps, like this recipe for a Bellini cocktail, a frozen variant. What you want, absolutely, to avoid is using a heavier sparkling wine; the Prosecco is light, and neither too acidic nor too sweet, and it doesn't overwhelm the flavor and scent of the fresh peaches.
Image credit: Sean Biehle