The reason that 에게 has so many different usages is that its function and meaning change with different kinds of verbs, which is why foreign learners of Korean are often confused by some of its usages. In fact, even some Koreans are confused.
Here are the ten usages that my grammar book ("외국인을 위한 한국어 문법2") lists for 에게. I have translated the explanations and the example sentences from the book into English.
- 에게 is used to indication the person or animal that will be influenced by the action.
* 제가 친구에게 책을 주었어요.
...I gave a book to my friend.
* 선생님은 학생들에게 노래와 춤을 가르쳤다.
...The teacher taught song and dance to the students.
* 어른에게는 존댓말을 써야 합니다.
...You should use polite speech with adults.
* 그 남자는 나에게 취미가 뭐냐고 물었어요.
...That man asked [to] me what my hobbies were.
* 동생이 친구에게 전화를 겁니다.
...My younger sister is calling [to] her friend.
* 닭에게 모이를 주었나?
...Have you given feed to the chickens?
- When used with such verbs as 가다, 오다, or their derivatives, 에게 indicates the direction or destination of the subject.
* 미영 씨는 왜 재훈 씨에게 왔습니까?
...Why did Mi-yeong come to see Jae-hun?
* 나는 머뭇머뭇 그에게 다가갔다.
...I approached [to] him hesitantly.
- When used with such verbs or adjectives as 있다, 없다, 남다, 많다, 적다, and 생기다, the marker 에게 indicates who is in possession of something that could be material or immaterial.
* 동생에게 급한 일이 생긴 것 같아요.
...It seems my sister is faced with an urgent matter.
* 여전히 저에게는 큰 문제가 남아 있습니다.
...I still have a difficult problem.
* 김 선생에게 그만한 돈이 있을까?
...Do you think Mr. Kim has that much money?
- When used with such verbs and adjectives as 맞다, 알맞다, 어울리다, 비하다, and 뒤지다, the marker 에게 is used to indicate the object to which something is being compared or measured.
* 그 옷은 저에게 너무 큽니다.
...Those clothes are to big for me.
*이 옷이 학생에게 어울린다고 생각하니?
...Do you think these clothes are suitable for a student?
* 이 사전이 학생들에게 알맞을 거예요.
...This dictionary is suitable for a student
* 동생은 키는 작지만 공부는 친구에게 뒤지지 않았지요.
...My younger sister is short, but in her studies, she keeps up with her friends.
- When used with such verbs as 잡히다, 빼앗기다, and 발견되다, the marker 에게 indicates who performs the action. (Notice that the verbs are passive verbs, so 에게 would normally be translated as "by.")
* 숨어 있던 도둑이 경찰에게 잡혔다.
...The thief that was in hiding has been caught by the police.
* 땅 속의 보물이 100년 만에 집주인에게 발견되었다.
...After being buried for 100 years, the teasure was discovered by the house owner.
* 친구에게 깜빡 속아서 일요일에 학교에 갔다.
...I was completely fooled by my friend and went to school on Sunday.
* 호랑이에게 물려 가도 정신만 차리면 살 수 있어.
...Even if you are being carried off by a tiger, you can survive if you keep your wits about you.
- When used with such verbs as 받다, 당하다, 얻다, and 배우다, the marker 에게 indicates the person who causes the action. 에게서 can also be used with these kinds of verbs. (I think 듣다 can also be included among these verbs.)
*아직 부모님에게 용돈을 받아요.
...I still get pocket money from my parents.
* 이 선생에게 한국말을 배웠지요.
Yes, I learned Korean from instructor Lee.
* 이 책을 친구에게 얻었다.
...I got this book from a friend.
* 저는 동료들에게 놀림을 받고 많이 울었어요.
...I cried a lot after being teased by my colleagues.
- 에게 is used to indicate the person who is instructed or made to do something.
* 엄마가 아이에게 제시간에 숙제를 마치게 했어.
...The mom made the child finish his homework on time.
* 여자는 누워 있는 남자에게 시원한 물을 마시게 했다.
...The woman had the reclining man drink some cool water.
* 선생님께서 내 짝에게 책을 읽히셨어요.
The teacher had my partner read the book.
- When used with verbs and adjectives that express feelings and evaluations, 에게 is used to indicate the person who is the subject of that feeling or is in the situation.
* 지금 딸에게 필요한 것은 아버지의 사랑입니다.
...What the daughter needs now is her father's love.
* 우리에게 참으로 귀한 것은 무엇일까?
...What is really precious to us?
*그 일이 너에게는 쉬울지 모르지만 그 아이에게는 매우 어렵다.
...It may be easy for you, but it is very difficult for that child.
- When used with verbs and adjectives that show emotion, 에게 is used to indicate the person who causes the emotion.
* 아주, 이제는 나 자신에게 실망했다.
...Damn it! Now I have disappointed myself.
* 여자는 상대방에게 호감을 느낀 듯했다.
...The woman seemed to show interest in the other person.
* 그에게 느끼는 감정이 사랑이야.
...The feeling I have for him is love.
- In the salutation of a letter, 에게 is used to indicate the receiver of the letter. (한테 is not used because 한테 is usually used in spoken Korean, not written Korean.)
* 사랑하는 벗에게
...To my loving friend
* 보고 싶은 동생에게
...To the younger brother I miss
...To my friend
Among the ten usages, using 에게 with the Number 6 usage seems the most strange to me, but since it says that 에게서 can also be used, I can learn to live with it. Also, I have not been using 에게 in the way shown in the Number 7 usage, but I will try to remember to do so in the future. As for the other usages, I do not seem to have much of a problem with them.
My grammar book says that 한테 can replace 에게 in all the different usages, except for Number 10. 한테 is usually used with spoken Korean, not written, so it would not normally be used in a letter. It also says that 에게 can be used with both spoken and written Korean.
My book also says that 더러 and 보고 can replace 에게 in usages Number 1 and Number 7. It also says that 에게로 can also be used with the Number 2 usage, but I do not like 에게로, for some reason, so will probably just be using 에게 in the case of usage Number 2.
Also, 나에게, 저에게, and 너에게 can be abbreviated to 내게, 제게, and 네게, respectively.