The most popular cake in Japan is probably ショートケーキ (shortcake), which is a type of layer cake. Usually a strawberry on top and in one of the layer with cream.
A good site to check how certain cakes look like or call at pâtisserie is here:
http://lesanneesfolles.jp/gateaux/
(my goto when I live in Ebisu)
Classic Cakes
ショートケーキ
Shortcake
キリッシュトルテ
Kirsch Torte
モンブラン
Mont Blanc aux marrons
A good one in Kyoto: http://www.malebranche.co.jp/montblanc/
Places
ケーキ屋
Any place sells cake.
パティスリー
A place sells cake and bakery. Originated from France and Belgium. The chef need to pass sort of certifications. But nowadays there are also places just claim/use this name as their shop name.