(Review participles first if these are unfamiliar.)
The genitive absolute is a clause that identifies a subject and action or state in which the subject of the clause is not involved in any way in the action or the state expressed in the main clause. It is called a "genitive absolute" because both the subject and the predicate (always a participle, not a finite verb, in this kind of clause) are both in the genitive case.
Contrast the two sentences below. In the first, the word 'boat,' which is the subject of the first clause, is not mentioned in the second clause (the main clause). In the second sentence, the 'boat' is referred to in both clauses, for it is the pronoun 'it' in the main clause.
In popular Koine Greek,
the strict rule that the subject of the genitive absolute cannot be referred
to the the main clause is often disregarded.