This is the 81st most frequent Italian word.
aver
An archaic or poetic form of "avere," meaning "to have."
In this sentence, 'aver' is used as the infinitive form of the verb 'avere' (to have), and it appears in an idiomatic expression meaning to pay attention or take care.
Aver cura dei dettagli è importante.
Having care for the details is important.
Here, 'aver' is part of the compound past tense, where 'aver' represents the auxiliary verb 'avere' in its reduced form followed by the participle.
Mi sembra di aver visitato questo posto prima.
It seems to me that I have visited this place before.
In this usage, 'aver' is required to construct a hypothetical past situation combined with modal verbs like 'potrebbe' (might).