What does "Anal. Calcd." stand for written out?
They are two distinct terms, that just happen to frequently be found next to each other. Their definitions can be found in the Journal of Organic Chemistry Guidelines for Authors (see page 29).

Anal. - abbreviation for combustion elemental analysis. 

Calcd - abbreviation for calculated. 

It is not actually a single phrase, as you've suggested here. Combustion elemental analysis is what people often mean when referring to 'getting an analytical done'. Writing Anal. signifies the type of experimental data that is about to be listed, just as you might specify 1H NMR, IR or MS. A common alternative to combustion elemental analysis is high resolution MS, which is often written in experimental sections as HRMS (ESI) m/z calcd 

The format for presenting elemental analysis is journal specific, and therefore varies. The format you're asking about is consistent with the ACS publication recommendations. RSC guidelines accept the the data in a slightly different format.