Slitherine

What is Field Training and how come I can repeatedly build it?

Those are operations, if you build one you are able to use it ones every few turns somewhere on the map of your choice. The more you build the more you can use simultaniously. they wil show in the left top of your screen after building them.

Some operations can be used every X turns, while some have limited number of uses.

General rule seems to be that if you build an operation, it's single-use (field training, cloning, etc), but if you build a structure that provides an operation (recon satellite, orbital bombardment, nuke, etc), then that structure keeps giving you that operation at regular intervals. Example: Nuclear Launch Center gives you a new nuclear missile operation every 10 turns.