"The old soldier is not fighting, but sitting and having a rest."
Translation:Az idős katona nem harcol, hanem ül és pihen.
Elderly and old. "Idős" is nicer to say.
Also, "idős" is almost exclusively used for people. "Öreg" can be used for other things, as well - probably for personalized objects and things.
There is one more word: "régi". It means "old", in the sense of "from old times". A typical example for that meaning is "an old friend of mine" - someone who can be young but be my friend since childhood.
"Öreg" is more commonly used and it is not insulting in my opinion. Like "Öreg néne őzikéje" in literature. :)