Europe has so many great places it's hard to narrow it down. Nonetheless, here's my shot at it:
Beautiful cities in general:
Lisbon, Portugal (pictured)
Budapest, Hungary
Valencia, Spain
Vienna, Austria
Poznan, Poland
Castles.
If you're into castles, Germany is a good place to explore. E.g. Neuschwastein Castle:
Other gorgeous places:
Cinque Terre, Italy
Santorini, Greece
If you're into hiking and 'lakes & mountains' scenery:
Switzerland and Austria would be hard to beat.
Beaches: lots of options on the Mediterranean.
Culture: some of the best museums, arts galleries, opera houses are in Europe (e.g. in Paris, London, Milan). Too many to list here and very much taste-dependent.
For more unusual scenery:
Head to Iceland. You can also see where the north american and eurasian plates come together in a national park there. Plus there's stuff like hot river springs and lagoons.
For a great drive:
Transfagarian road in Romania
If you're after a place with a rich history/backstory:
Istanbul, Turkey
Nicosia, Cyprus
Politics aside, the island is divided into the 'Cypriot' part and the 'Turkish' part. Nicosia is a divided capital, the UN buffer zone runs through the city.
Tiraspol, Transnistria
Breakaway territory from Moldova. It's like a country 'frozen' in time since the Soviet days. Officially this place doesn't exist (it wouldn't usually be shown on maps as it's not recognized as a country). Be prepared to pay bribes here for bullshit reasons and other hassle.
Chernobyl, Ukraine
I think you can still go visit the reactor area and the nearby
abandoned town of Pripyat (pictured) on a tour.