Historic buildings, the Royal Palace, Fisherman's Bastion and some very good restaurants in that area. Too many little souvenir shops. Some very interesting statues are in the area, a tram (number 16 and 16 a, I believe) that goes through the area is easy to use if you have the tickets bought ahead of time....a few days in Budapest will allow you to know how to use the trams which are convenient for traveling longer distances, and going from Buda to Pest more easily. The old Matthias Church was being renovated when we were there, the crowds were thick so it was frustrating, but it is definitely worth a visit.
You can take the funicular down to the Danube, and cross the Chain Bridge to go to Pest, where there is a huge market inside a building, with excellent rates on their paprika, tons of food items and great souvenir gifts.
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