The Voice of All Things refers to the words conveyed by inanimate objects and animals that don't speak the human language. Although these words are usually either inaudible or unintelligible to humans, there exists an innate ability that allows a person to perceive these words and to understand and gain information through them. For example, it is possible to use this ability to gain information from the Poneglyphs, as well as track them down.
Some people are able to converse with the voice and, in case it belongs to an animal, might even be able to command it. However, those who are able to perceive the Voice of All Things are usually unaware that they possess the ability, and don't even know about the Voice of All Things itself.
Furthermore, as shown with Roger, though he was able to hear the Poneglyphs voice, this did not mean he could read the text or decipher its meaning, he simply understood that it held great importance.
Kozuki Oden was also able to hear the voices of the Sea Kings and the great voice at Zou similar to Roger.
Monkey D. Luffy is able to perceive and understand animals like the Sea Kings and Zunesha as well, but has not shown the ability to perceive the voice of non-living things.
Kozuki Momonosuke, Oden's son, is also able to perceive Zunesha. Similar to his father and Roger, he felt strange after arriving at Zou, which got stronger when he got closer to the Whale Tree. Unlike all others, Momonosuke was actually able to converse with the elephant, giving it the order to attack the Beast Pirates.
The Three-Eye Tribe is rumored to have access to this ability through utilizing the power of their third eye. However, mixed-blood members might not necessarily be able to use it at all (or at least, implied by Charlotte Linlin, it could awaken later in life).