Budapest used to be three cities, Buda, Obuda and Pest, and nowadays is a big city to explore. The first morning I made the mistake to try and cover it all on foot, but by using the effective underground and streetcar system you can spare your feet.
The Castle Quarter in Buda is the most touristy part. It has the usual overpriced restaurants, but also some pretty sights. It wasn't too crowded, nothing like the madness I experienced in Prague or Florence.
On the other side of the Danube, in Pest, more recent monuments can be found. The Parliament looks a lot like its English counterpart.
Most of the buildings along Andrássy Avenue also date from the 19th century.
I couldn't find exact information on which of the many monuments in Budapest's city center are included in this World Heritage Site. The oldest underground railway on the European continent is (the oldest in the world, London, however isn't!), but the largest Synagogue in Europe is not (as far as I can make out). There are too many interesting places to choose from in this city well worth visiting, and by far I didn't see them all during my 3-day stay.