The Voice of All Things refers to the words conveyed by inanimate objects and animals that do not 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 do not even know about the Voice of All Things itself.
Furthermore, as shown with Roger, though he was able to hear the Poneglyphs' voice (enough to have a vague idea of its contents), this did not mean he could read the text or decipher its meaning, he simply understood that it held great importance.
It seems that sentient beings are able to communicate their will across great distances using the Voice of All Things. During the Fire Festival, despite being unconscious and drowning, Luffy was able to use the Voice to communicate with Momonosuke who was on a floating island high above the ocean's surface, and had him pass on a message for him. It also seems the Voice can be heard by those less sensitive to it if they are close by, as Luffy was able to call the Heart Pirates to save him while he was drowning.
Kozuki Oden: Oden was also able to hear the voices of the Sea Kings and the great voice at Zou similar to Roger.
Monkey D. Luffy: 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. Luffy is also currently the only known human to, both, be able to hear the Voice and use it to communicate as well.
Kozuki Momonosuke: 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-Eyed Tribe: The members of the Three-Eyed Tribe are 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).
Zoro displayed a similar ability during is fight with Daz Bonez. While being close to death, Zoro was able to feel the presence and souls of the rocks that were about to crush him, which allowed him to dodge them. He could also feel the soul of his sword which allowed him to find them under the rubble created by Daz' attack.