音訓 (음훈) or 訓音 (훈음) is short for 音讀 (음독) + 訓讀 (훈독), referring to the short dictionary gloss given to Chinese characters by Korean dictionaries. It is typically formatted in the following order:
國 (漢字) 나라 (訓讀) 국 (音讀)
where 訓讀 is a Korean translation (usually, but not always, using pure Korean words) of the character, and 音讀 is either the widely accepted Chinese-derived reading in Korean or a fanqie reconstruction for rarer characters.
Similar formatting is still widely employed in Japanese, and formerly employed in Vietnamese dictionaries of Chinese characters.
And why don't we say then 의미 then, why 훈?
訓讀 are the ordained, official translations for the dictionary gloss system of 音訓. There are countless ways of translating any Chinese character into Korean with any variable number of words (any one of which could be considered an "의미" of the character), but dictionaries only prescribe a restrictive reference set and call this set "訓讀".
Basically, 訓讀 is a traditional system purely used for Chinese character dictionaries. It is easy to see that such rigidity was necessary in older times, when translating Classical Chinese documents were crucial and expected to be consistent, and this system has simply survived to the modern age, regardless of its practicality nowadays.