This is the 10352nd most frequent Russian word.
бывшему
"To the former" or "to an ex."
Here, 'бывшему' is in dative case, meaning 'former' and describing 'friend' who is no longer in that role.
Я написал письмо бывшему другу.
I wrote a letter to my former friend.
In this sentence, 'бывшему' is once again in dative case, describing 'neighbor' who previously had this relationship with the speaker.
Он подарил подарок бывшему соседу.
He gifted a present to his former neighbor.
Here, 'бывшему' modifies 'colleague', indicating that the individual used to be in that position but is not anymore.