This is the 668th most frequent Icelandic word.
býr
"Lives" or "resides".
In this sentence, 'býr' is the 3rd person singular present tense form of the verb 'búa', meaning 'to live (reside) or dwell'. It indicates a location in which someone resides.
Hann býr í Reykjavík.
He lives in Reykjavík.
Here, 'býr' again is used in the context of residing, describing the location where the family resides.
Fjölskyldan okkar býr á Íslandi.
Our family lives in Iceland.
In this case, 'býr' is used in a different context; 'búa til' means 'to make/create', so 'býr' is part of a phrasal verb to indicate creation or production.