You'll find Tihany on the northern shore of Lake Balaton (Hungary) on the peninsula with the same name.
The center of it is the Benedictine abbey founded in 1055 by András I, who is buried in the crypt. It is a bit of a tourist magnet, because of its looks and its view over Lake Balaton. For people interested in history it's most important because its founding charter is the oldest remaining record of Hungarian language. That charter however is held at the Pannonhalma Benedictine Archabbey. And for those interested in petite histoire, the nice little details of history: the last Habsburg emperor of Austria, Charles I, was briefly held prisoner here following his second attempt to regain the throne of Hungary.
Tihany is also famous for an echo that is no longer there due to changes in the landscape and goats' nails washing on the shore, which are actually molluscs. Both stories are "historically" connected to a very proud and hard hearted princess which got cursed by the king of the lake. Tihany's inhabitants did not get cursed however and have the highest per capita income in Hungary (and the highest housing prices).
Oh yeah, the pensinsula used to be very well known for its lavender fields (of which the oil was apparently of better quality than the French), and you'll still find lavender for sale all over the island, but you'll not stumble over the lavender wherever you go anymore ...