Which database of celestial objects do stargazing apps use?
The Hipparcos is probably all that's really needed.  A lot of them add in portions of the Tycho catalog as well.  But even the Hipparcos catalog has more stars than what most armatures will need, and it's generally impractical memory wise to use the whole thing.  To really go crazy, a torrent for the NOMAD catalog is available.  All of those list only individual stars.  For things like galaxies and nebula the NCG catalog should do, but there are also the Index Catalogue, and Hershel Catalogs.  The most popular catalog would be the Messier, which has no official distribution sources (that I know of), but databases are available if you search this one for example.

If you don't care about the original sources, Stellarium has their star catalogs on their wiki.