This is the 884th most frequent Icelandic word.
ingibjörg
"Ingibjörg" is an Icelandic female given name, meaning "protected by Ing" (Ing being a Norse god).
'Ingibjörg' here is again a proper noun, representing a person, likely as the subject of the sentence.
Ingibjörg kennir íslensku í skólanum.
Ingibjörg teaches Icelandic at school.
'Ingibjörg' is used as a proper noun, being the name of a person.
Hún heitir Ingibjörg og er góður vinur minn.
Her name is Ingibjörg, and she is a good friend of mine.
'Ingibjörg' is used in the accusative case, 'Ingibjörgu', indicating the direct object of the action.