There are many beautiful old synagogues in Budapest, in fact, one of the biggest in the world is found here. I knew most in the Jewish Quarter, one in Zugló and another in Óbuda, and visited one a few times as an anthropolgy student near Keleti Station. Based on information found on the internet, next is a comprehensive list of the Budapest synagogues with photos to be added along the way.
Dohány Street Synagogue
(the main synagogue)
Kazinczy Street Synagogue (Sasz-Chevra Orthodox Grand Synagogue)
Rumbach Street Synagogue
Frankel Leo Street Synagogue
Front

Alef Kids - Óbuda Synagogue
Óbuda, former synagogue
Laborc street
Dózsa György Street Synagogue
Thököly Street Synagogue
Bethlen Square Status-Quo Ante Synagogue
Dob Street Kosher Butcher
Medieval Jewish Prayer House
Other synagogues in Budapest
Dessewffy Street synagogue
Teleki tér shul
Páva utcai Zsinagóga
Újpest Synagogue
Vasvári Pál Street Synagogue
Shas Chevra Lubavitch synagogue
Congregation Dor Hadash
Keren Or Cultural Center and House of Prayer
Izraelita Egyház zuglói körzet imaháza
Pesti Jesiva - Zsidó Oktatási Központ