The Voice of All Things refers to messages conveyed by inanimate objects and animals that do not speak the human language. Only a very small number of people in the world possess the ability to hear this voice.
The Voice of All Things is not an ability which simply entails understanding animal communication. Rather, the messages that come from these animals (in addition to objects) appear to be completely inaudible to anyone who cannot hear the Voice; thus, the communication can be considered akin to telepathy. To this point, it does not appear as though this is a universal trait among every single object and species; in fact, everything that has communicated using this voice has been an animal or object of quite notable status in the world: the powerful Sea Kings, the giant millenium-old elephant Zunesha, and the Poneglyphs. This attribute was noted by Gol D. Roger, who came to understand the Poneglyphs' great importance due to being able to hear their voice. Interestingly, the Poneglyphs were developed by the Kozuki Family, whose two most recent leaders are known to have been able to hear the Voice.
So far, only a minuscule number of people in the world have been shown with the ability to hear the Voice. Only one race in the world, the Three-Eye Tribe, possesses the innate ability to hear it; not only is this race extremely rare, but its members must also work to awaken the ability. In most instances, whenever someone with the hearing ability comes in close contact with an animal that can use the Voice, they will hear a constant, seemingly unintelligible noise, which can overwhelm them especially if they are unfamiliar with the concept of the Voice and the ability to hear it.
It also appears as though people can hear this voice to different degrees. Whereas Roger, Oden, and Momonosuke constantly heard the Voice or at least "felt" it around the Sea Kings and Zunesha, Luffy showed no indication of this; however, the Sea Kings still noted that Luffy had picked up their voices, and Luffy was finally able to hear Zunesha when it called out for help. Additionally, there exists an ability for the hearer of this voice to communicate with the source, as well as see through the source's eyes. This was seen with Momonosuke and Zunesha, and is also not a universal ability, as Luffy was unable to perform this despite hearing the creature.
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. Big Mom perceives that the Voice of All Things will enable one to decipher the Poneglyphs, but given Roger's observations, whether or not this belief is true is uncertain.
During the Fire Festival, despite being unconscious and drowning, Luffy was able 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. While this was significantly different application compared to previous usage of the Voice of All Things, it possessed essentially the same elements and effect. More inexplicably, Luffy was also able to call to 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.