I think one of the most popular cocktail that starts with A is the Appletini or Apple Martini, the basic recipe calls for -
3 parts vodka
1 part apple schnapps
1 part cointreau
Shake the ingredients together in a cocktail shaker with ice and strain to serve in a cocktail glass like this one - this is a fun way to drink appletinis in a glass made specially for them!
Of course there are several other versions of the appletini available and you can find recipes for the sour apple martini and extreme apple martini amongst others at Drink Recipes: Appletini or Apple Martini.
Of course there are far more cocktails that begin with the letter a than just appletinis as you will know if you've already clicked on the first link. One which I think is a great cocktail that most people won't have tried before is the Asian Pear Martini, this is a very modern cocktail and is sure to be a hit when you serve it.
Asian Pear Martini
2 shots sake
1 shot pear liqueur
1 1/2 shots fresh pear juice
1/4 shot fresh lime juice
Pop all of the ingredients into a cocktail shaker with ice and shake well before straining into a cocktail glass to serve.
Asian style cocktails are very in vogue at the moment and you can get a range of different types from the Asian Pear Martini here to some very exotic Vietnamese and Indian cocktails. For more information on Asian Inspired Cocktails I do have a resource for you to check out that is aptly called Asian Inspired Cocktails!