An absolute phrase, or a nominative absolute, is generally --but not always--a group of words consisting of a noun/ pronoun and a participle as well as any related modifiers.
It is one more helpful linking device.
For example, "His voice was trembling and his eyes were roving" and "He started to speak" can be joined as "His voice trembling and his eyes roving, he started to speak", where "His voice trembling and his eyes roving" - is an absolute phrase. The absolute phrase modifies the whole sentence rather than a particular word. It adds up new details to the general meaning of the sentence. Absolute phrases are regarded as parentheses and they are separated from the rest of the sentence either with a comma (pair of commas), or a dash (pair of dashes).
If the participle of an absolute phrase is expressed by the verb to be ( e.g. being, having been ), it is often omitted, but can be easily understood.
Examples: The party (being) in full swing, she decided to leave.
(Having been) married for twenty years, they could not imagine their lives without each other.