FIRs and UIRs are, in effect, administrative boundaries. Smaller states will have only one FIR, larger states break their airspace up into several.
FIRs do not necessarily correspond to the extent of airspace controlled by an ATCC (center). The US chooses to break its airspace up by center for IFR purposes, but it's not quite the same as an FIR.
What changes as you move from one to another? For an IFR flight in controlled airspace, not much. You might anticipate a frequency change, particularly at an international boundary, but there are plenty of frequency changes between sectors within an FIR anyway.
UIRs are just FIRs at a higher level, and the tracts of airspace are often bigger, with one UIR overlying several FIRs.
A CTA is a control area, as already stated. The continental US is one big CTA from 1200 agl up to FL180, with local exceptions of course. It used to be called the "continental control area" I think, before alphabet soup days. Being a CTA doesn't make it a particular class of airspace, except that you know that it must be class A to E and that IFR flights will receive a control service.