A common phrase/word I've seen used in text and I've heard on air in conversation is "hihi" or "hi-hi". What does this mean and in what contexts is it appropriate to use?
It is ham radio laughter. HIHI, sometimes HI, other other times HIHI. It's origins are in CW (aka Morse Code), not voice. In fact, I believe old-timers might think it is silly to say HiHi or something on SSB or other voice comms when you can merely laugh if something is funny.
But, with Morse Code, laughter is not in the alphabet so HIHI gets the job.
In Morse code, this is "di-di-di-dit di-dit" -- and the pattern is supposed to vaguely sound like laughter (I think very vaguely). I actually never use it even though I have been mostly CW ops for decades and decades.