I don't mean to offend the other answerer but her answer is partly correct. There is no basic difference between 잊다 and 잊어 버리다 in the meaning. Both mean "forget". However, when a verb is followed by an auxiliary ending particle of "버리다", the verb can have an additional nuance, which means the verb's action is completely finished. In a way, I think it is similar to present perfect tense in English. Anyway, when 버리다 is added to a verb, the completion of the verb's action is emphasized and sometimes it can also imply that the speaker feels sorry or relieved because it is already finished. Please see the following examples. 1) 그녀의 전화번호를 잊었다. I forgot her phone number. This sentence simply indicates the fact that I can't remember her phone number. 2) 그녀의 전화번호를 잊어 버렸다. I have forgotten her phone number. This sentence is stronger than 1) and emphasizes the accomplishment of forgetting. It could also express the speaker's anxiousness that he can't call her. 1) 숙제를 다 했다. I finished my homework. 2) 숙제를 다 해 버렸다. I have finished homework. Again, this sentence emphasizes the completion of doing homework and could imply that the speaker feels relieved thereby.

잊다 and 잊어버리다 They technically mean the same thing because 버리다 is an auxiliary verb (helping verb). It means 'to throw away'. So, 잊다 'to forget' + 버리다 'to throw away' === 'forget it and throw away!' So, 잊어버렸다 more emphasizes that 'forgot'

잊다 is used if the doer of 잊다 deliberately forgets something.[잊다는 '잊다'의 행위자가 고의로 무엇을 잊으면 사용돼.] 잊었다(잊-+었다) is the past verb of 잊다.[잊었다(잊-+었다)는 잊다의 과거동사야.] 잊어버리다 is used for indicating the doer forgets something unintentionally.[잊어버리다는 행위자가 무심결에 무언가를 잊어버렸다는걸 나타내기 위해 사용돼.] Similarly as making past verb of 잊다, you can make past verb of 잊어버리다 like this way.[ 잊다의 과거동사와 마찬가지로 '잊다'의 과거동사를 이런 방식으로 만들 수 있어.] :'잊어버리-+었+다'='잊어버렸+다'='잊어버렸다' 나는 지난날의 기억들을 다 잊었다.[I forgot memories of days in the past.] 나는 도서관에 들러서 책을 반납하는 것을 잊어버렸다.[I've forgot to stop by library to return books.]

Lol... who knew this question would be so controversial 아이고~