What pastry and chocolate are to Western confectionery, mochi and red bean paste are to Japanese confectionery. I'd say more, but this page needs some work.
Manju - A wheat bun which is filled with something and then steamed. If you do dim sum, you're already familiar with these as chrysanthemum buns. The only proper, good, and rightful filling for manju is fork-roasted pork, whereupon the bun becomes a roast pork bun, otherwise known as the Food of the Gods. However, buns filled with red bean paste (like the one Tasuki used to bribe Miaka) are hereby permitted.
Manju come in several sizes, ranging from ones as big as your fist to teeny-tiny ones barely an inch and a half across which come in microwavable packages. If you're out of reach of a Chinese or Japanese bakery, those microwaveable packages are your best hope of getting manju. The freezer section of an Asian grocery is the best place to check, obviously, but your supermarket freezer might surprise you.
Crackers - The Japanese make piles and piles of sweet crackers. I can't call them cookies because they're not; they're ordinary, unsweetened crackers brushed with a sweet coating. Often their unsweetened look-alikes are sold on the shelf right next to them. The usual sweeteners are sugar syrup, mirin, and a mixture of mirin and soy sauce which makes for a truly... unique sweet-and-salty flavor.
Cookies - Of course, they make cookies too. The large ones tend to be expensive, fancy, and either very sweet or only mildly sweet, with plenty of cream filling; the small ones, which look a lot like goldfish crackers or Combos, are much less expensive, more lowbrow, and usually fairly sweet, with plenty of cream filling. The fillings are usually part of the holy trinity of vanilla-chocolate-strawberry.
Cake - The Japanese looooove cake. They even have a special holiday devoted to the eating of cake, Kurisumasu. (Does this make "kurisuu" the verb for "to eat cake"?) Since cake is a Western import, the cakes are pretty much what you'd expect to see.
I haven't seen any whole Japanese cakes in the U.S., possibly because American cakes do just fine. However, Asian groceries do sell slices of pound cake and jelly or cream roll, usually in the refrigerated section. I haven't tried any, but if you have, write to me and tell me how it was.
Ice cream - Mmm, red bean ice cream. Green tea and azuki (red bean) ice cream is widely sold in the States. I've had azuki ice cream, which was delicious and reminded me faintly of strawberry ice cream.
If you can't find Japanese ice cream in any of the usual places, try one of the boxed ice cream mixes. Making them is fairly simple--usually a matter of adding milk and freezing.
Purin - Creme caramel shots! I kid you not. Teeny-weeny servings of creme caramel in little foil-topped cups; you can scoop the purin out with a spoon, or if you lack couth, you can throw your head back and suck the purin straight out of the cup. Highly recommended.
Pocky - Biscuit sticks dipped in chocolate. The archetypal Japanese sweet snack food. Comes in at least a dozen flavors and variations: milk chocolate, white chocolate, strawberry, almond, white chocolate with almonds, milk chocolate with almonds, milk chocolate with coconut, creme brulee, giant pocky... Apparently the dark chocolate pocky wasn't selling well enough, so they gave it a dark green box and a new name: Men's Pocky. Say it with me now: *deep, manly voice* MEN'S POCKY! Several pocky imitators have sprung up, chief among them Pretz. Pretz has thrown off the yoke of chocolate and expanded into new, non-sweetened frontiers, like... tomato. For more information about Pocky and its imitators, check out the Pocky pages at Japan for the Uninvited and Metropolis: Big in Japan., or buy your own Pocky from J-Box.