What are the Purposes of the Types of Meringue that are used for Pastry?

great question.

first of all, each meringue originated from a different culture and as such is more prevalent within recipes of that culture. more modern recipes might choose to use a different variant of meringue to get a more nuanced texture than in classic recipes.

I would like to refer you to a great article about this issue. here is an excerpt:

French:

sometimes referred to as "ordinary"—is the most basic of the trio and the least stable until baked. Egg whites are beaten until they coagulate and form soft peaks, at which point sugar is slowly incorporated until the mixture has attained full volume; is soft, airy, and light; and stands at attention when the whip is lifted. French meringue is customarily spooned or piped into different forms, including dessert shells (such as vacherins) and cake layers (as in a dacquoise), and baked, later to be topped with fruit, mousse, or whipped cream. It is also often folded into batters (for lady fingers, sponge cakes, soufflés, and the like) and baked.

Swiss:

is prepared by gently beating egg whites and sugar in a pan that sits above boiling water, without touching it. When the mixture reaches 120 to 130 degrees Fahrenheit and the sugar is completely dissolved, the mixture is pulled off the heat and beaten vigorously to increase and attain full volume and then at a lower speed until cool and very stiff. Swiss meringue is smoother, silkier, and somewhat denser than French meringue and is often used as a base for buttercream frostings.

Italian:

is made by drizzling 240-degree Fahrenheit sugar syrup into whites that have already been whipped to hold firm peaks. Whipping continues until the meringue is fully voluminous, satiny, stiff, and cool. Italian meringue is often used to frost cakes (alone or as a base for buttercream frostings), to top filled pies, or to lighten ice creams, sorbets, and mousses.