What is the meaning of ´olsun´ in English? I hear this word used in different ways. Getmiş olsun. Afiyet olsun. But it seems to mean different things in each of these. 

it is imperative form of an auxiliary verb to be (olmak )  but in these kind of phrases it has different meaning. it means  to wish

Afiyet olsun ( enjoy your meal )

Geçmiş olsun ( get well soon )

Uğurlar olsun ( have a good trip )

.....

and another use for you

A: Geç kaldım. Üzgünüm. ( i am late,  sorry )

B: Olsun. ( O.K., Fine! Alright! it doesnt matter )

Thanks! Now I understand. Makes much more sense to me now.

OLSUN means "let it be"

Afiyet olsun Let it be good health to you ! (to a person eating)

Geçmiş olsun Let it be over (finished) ! (to a sick person)

Uğurlar olsun Let it have good (beneficial) results ! (to a person going on a voyage)