What do all the numbers and letters mean in celestial bodies?

 i.e. APM 08279+5255 [Super massive black hole]

They are catalog numbers. Usually the letters at the beginning identify which catalog the object is listed in, the numbers identify the object within that particular catalog. The same object may appear in different catalogs under different numbers.

Particularly, the APM in your title is the Automated Plate Measuring Machine at the Institute for Astronomy, Cambridge. http://sundog.stsci.edu/first/APM/HTML/node3.html . The quasar APM 08279+5255 is also known as IRAS F08279+5255 and QSO B0827+5255. https://en.wikipedia.org/wiki/APM_08279%2B5255

The most well known catalogues are M (Messier) and NGC (New General Catalog). Long story short, Messier wanted to immortalize his surname by discovering a comet, so in order to avoid confusion he started surveying the skies and writing down all objects that look like a comet but are not (i.e. all diffuse objects like nebulas or galaxies). He did discover several comets, but became famous for his very complete catalog instead.

For other catalogues see: https://en.wikipedia.org/wiki/List_of_astronomical_catalogues

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