堅い (かたい) is a word I had a basic feel for long before I knew the actual English equivalent. I took an intensive summer class after my first year in university, and I have memories of the sensei using it all the time and me not exactly understanding what it meant when they used it to describe different words and phrases. I probably looked it up once and then just let it sink in. Unfortunately I can’t think up any specific examples of what was and wasn’t 堅い. It might have just been keigo variants.
Needless to say, having an understanding of 堅い is immensely important in Japan. The rigidity of your speech, your body language, your overall interactions with other people – these are all very important in Japan. Arguably more important than elsewhere. This isn’t to say that you should tense up like a plank when you talk with your 社長 (all 社長 should be lovingly referred to, behind their backs, as “the Shach” – let’s make it happen, y’all!). The most respected ability in Japan is being comfortable with your situation and knowing when and when not to dial up the intensity/formality/rigidity/堅さ.
Because this is such an important idea, it can be abused for humorous reasons, as I’ve mentioned a number of different times – most recently in my article in the Japan Times today about universal humor. Check it out!