알아듣다: catch, get, comprehend
Definition 1: to hear what someone says and understand what it means.
Definition 2: to hear a certain sound and know what it is or who made it.
EXAMPLE SENTENCES
First meaning
그가 내 농담을 알아들은 것 같지 않아요.
I don't think he got [understood] my joke.
그녀의 영어는 악센트가 강하지만 알아들을 수 있었어요.
Her English was strongly accented but intelligible.
미안하지만, 무슨 말씀인지 알아듣지 못했습니다
I'm sorry, I didn't catch what you said.
Second meaning
미안해요, 당신 목소리를 못 알아들었어요.
Sorry, I didn't recognize your voice.
목소리는 들렸지만 뭐라고 하는지 알아들을 수 없었어요.
I could hear voices, but I couldn't make out what they were saying.
태어난 지 반 년 정도가 지나면 아이는 엄마와 아빠의 목소리를 알아듣는다.
About half a year after birth, the child understands the voices of the mother & father.
이해하다: understand, comprehend, figure out
Definition 1: to realize and know something, to know something well and accept it.
Definition 2: to know and comprehend someone's situation.
Definition 3: to know what somethung is, to accept something as a certain thing .
EXAMPLE SENTENCES
First meaning:
지수는 수학 공부를 하는데 문제를 이해하지 못해서 풀 수가 없었어요. Jisoo couldn't solve the problem because she didn't understand the problem while studying math.
나는 편지를 읽었지만 내용을 이해할 수 없었어 I read the letter, but I couldn't understand it.
Second meaning:
경찰은 빵을 훔칠 수밖에 없었던 사정을 이해하고 적당히 훈계한 후 그를 놓아주기로 했어요.
The police understood the circumstances in which they had no choice but to steal the bread, gave him proper admonition, and decided to let him.
제가 사정은 이해하지만 지금으로서는 해 드릴 수 있는 게 없네요.
I understand the situation, but there's nothing I can do for you right now.
Third meaning:
저는 이 작품을 단순한 저작이 아닌 하나의 철학으로 이해하려고 해요.
I try to understand this work as a philosophy, not just a work.
그는 자연을 있는 그대로 보는 것이 자연을 올바르게 이해하는 방법이라고 말했습니다.
He said that seeing nature as it is is a way to understand nature correctly.