State
Historically
Before computers, before even the modern concept of a political state, state often meant 'condition' or 'position', and for a time often referred to social and political standing.
The origin of state in the political sense seems to come from the middle ages, when some analysis that was looking at different angles on local organisation areas, and ways to rule them, needed a term to generalize over, meaning something abstract like governed political unit, and used 'state' for that. (verify)(verify)
In this sense states are mostly territorial, administrative, and/or coercive organisations.
Weber's view is also interesting, pointing out it's any community that has claimed monopoly on the legitimate use of physical force within a given territory, for as long as that is seen by its people as legtimate, and the way they protect it with bureaucracy and such.
Engineering
Around engineering, modelling, and programming, state usually means "information relevant to its behaviour".
Sometimes pointing at some specific core details -- and then frequently a snapshot of that in time -- and sometimes in a more general sense of 'whatever is relevant' or even 'whatever is there'.
When engineering and modeling, we often end up focusing on a minimal set, whatever is enough to explain/model the specific behaviour/purpose we care about.
In programming, state usually refers specifically to whatever variables are relevant to behaviour (sometimes all that are kept, sometimes just the ones most relevant to behaviour, but aside from superfluous variables there is not a lot of difference).
Stateful and stateless
Applies most often to programming,
- Stateful tends to be mean "has to keep remembering something (often per case) to function",
- Stateless then means "does not need to rememeber anything specific to keep functioning"