Kyle can choose the shirt he’ll wear for the day at random—they’re all orange. This sort of use of “at random” to mean “by chance,” is perfectly standard. (Kyle should get some new shirts, though.)
Less widely accepted are a couple of slangy uses of the word, mostly by young people. In the first, “random” means “unknown,” “unidentified” as in “some random guy told me at the party that I reminded him of his old girlfriend.”
The other is to use random to mean “weird,” “strange,” as in “The party at Jessica’s was so random, not what I was expecting at all!” Evidently in this expression randomness is being narrowed down to unlikelihood and that is in turn being connected with strangeness, though randomness in real life is usually quite ordinary and boring.
Use of either of these two expressions in formal speech or writing is likely to annoy or confuse your audience.