Kitcho (吉兆) is probably the most famous restaurant for traditional Kyoto cuisine (also known as kaiseki-ryori 懐石料理). They have three restaurants in Kyoto: 1) the main historical location in Arashiyama, where there is no menu, only a price tag of about 50000円 per person, drinks not included of course, and since they do have a few bottles of Romanee-Conti '85 (yes, they do), the bill could be quite a bit steeper, 2) a place in the remote south-west of Kyoto, on the grounds of the Shoka-do garden and finally 3) the 'Hana' Kitcho, located in the center of Gion, 30m from the busy Shijo street.
The latter is probably the one you want to try if you wish to sample the amazing kaiseki-ryori. It is less formal (although still quite a bit!), cheaper (with prices from sub-10000円 to 30000円) and better located than the others. The website has a nice form in English but this is still a very traditional place so don't expect them to speak fluent English. Not a big deal since you have to reserve your menu a couple of days in advance anyway, so that the only works you may need are "beer", "sake", etc.
If you have a look at the website, you will see that the menus are specified with the number of items in them. That's because one aspect of kaiseki is that dishes are brought one by one, hence it's indeed countable. But it's a bit misleading, as one can quickly calculate that the extra dish between the 'Okina' and 'Kisashi' courses leads to a 50% premium! Obviously, the dishes are not the same and the most expensive menu will have much more extravagant food.
You can book a special room or eat in the main dining room; I would recommend the private room if you have a few guests: they are in the classic sukiya style and offer a better atmosphere. There's a premium for private rooms too. And there's also some extra tax if your bill gets above a certain amount (forgot how much) so the bill can climb pretty quickly. But it's good. Really good.
Nearby: Humorous shrine 壹錢大明神 (60m), Shirakawa 白川町 (190m), Shōdeneigen-in 正伝永源院 (200m), Pontochō 先斗町 (210m), Daichū-in 大中院 (320m), Kennin-ji 建仁寺 (360m), Seirai-in 西来院 (380m), Kyūshō-in 久昌院 (380m), Ryōsoku-in 兩足院 (410m), Zenkyo-an 禅居庵 (450m),...
External links: Website, ウィキペディア, Kyoto Navi, Satellite view, Map
Keywords: Japan, 日本, Japon, Kyoto, 京都, 東山, Higashiyama, 花吉兆, Hana Kicho, kaiseki, 懐石料理, restaurant, レストラン