N1, N2 and N3 refer to the engine spool speeds. N1 refers to the low pressure spool, N2 refers to the intermediate pressure spool and N3 refers to the high pressure spool. An engine with only one spool will use N1, an engine with 2 spools will use N1 and N2. An engine with three spools will use N1, N2 and N3. The fact that an engine has two or three spools does not necessarily mean that all of the N values will be indicated to the pilot.
In some cases N1 is used as an indication of thrust, but this is not always the case.
In the case of free power turbine engines the hot high-energy gas is produced by the gas generator and then fed through the free power turbine. In such cases the gas generator speed may be referred to as NG (G being for gas generator) or NC (C being for compressor). But there is no reason why N1 or possibly N2 could not be used.
Free power turbine speed is referred to as NF (F being for free turbine) or in some cases NT (T being for turbine)
NR is commonly used to refer to the main helicopter rotor speed.
Over the years different manufacturers have used different systems, so there is no fixed system that can be said to apply to all aircraft/engine combinations.