This ornate reliquary houses the mummified hand of St. Stephen (975-1038), the first crowned king of Hungary. The relic is still venerated in remembrance of Stephen's conversion of the Magyars and unification of Hungary. Curious Expeditions has the full story and more pictures.
An interesting side note regarding the crown of St. Stephen, from Wikipedia:
The Holy Crown, popularly attributed to St. Stephen, was removed from the country in 1945 for safekeeping, and entrusted to the UnitedStates government. It was kept in a vault at Fort Knox until 1978, when it was returned to the nation by order of President Jimmy Carter. It has been enshrined in the Hungarian Parliament building in Budapest since 2000.
At the main alter of the Basilica is also pieces of the leg bones of St Gellert as well!