This might be a bit long~
Shiki has a very unique and special ability. By experiencing death once, his mind and soul became linked with Akasha, the Spiral of Origin that records all phenomena in existence, whether it be past, present or future. Since Shiki was already born with innate psychic abilities, this opened up a channel within Shiki's brain, allowing to see "death" in the form of lines and points. This ability is called the Mystic Eyes of Death Perception ("Chokushi no Magan"). By tracing along the lines with a sharp enough object, anything, whether it be a bed, a chair, human flesh, can be cut apart as easily as wet tissue paper. By piercing the point of an object, that object will cease to exist, thereby "stopping" it, and as a consequence of no longer existng, bringing it to "death". In other words, what Shiki sees is the predetermined "death" of an object. As long as it exists, no matter who or what it is, Shiki can "kill" it by tracing the lines or piercing the points. Depending on the user's level of ability, it may even be possible to "kill" concepts and phenomena, although this Shiki hasn't quite reached such level yet.
This ability also ignores any and all defences, as well as regenerative ability. No kind of healing or regeneration, whether it be natural healing, supernatural healing, or even spatial manipulation can mend something cut by the user. The only known cases of someone "healing" a wound inflicted by this ability are: Arcueid Brunestud, who had to reconfigure her entire conceptual structure (Tsukihime), and Araya Souren, who had become a semi-conceptual existence (Kara no Kyoukai).
It is, however, possible to be immune to this ability. Shiki can only kill beings his mind is can comprehend and considers killable. Other humans, animals, plants, non-living things like rocks and air, and even things that are already dead. Everything has the same "root", and thus have similar concepts of death. As long as the origin can be traced to the Earth, no matter what it is, Shiki will have the subconscious knowledge necessary for him to comprehend its end. This most likely also applies to being not necessarily from earth, but have fundamental similarities to beings to them. For example, on the Island, there are various beings who, while human, aren't from the same planet as Shiki is. Shiki can probably see the lines of these beings, as they have few fundamental differences from their earthly counterparts.
Conversely, Shiki cannot see the death of things that don't have a concept of death that he can understand. The best example of this is ORT, the ultimate being from the planet Mercury. The reason Shiki cannot kill ORT isn't because it's from another planet, but because its existence is far too removed from his own. His mind cannot comprehend ORT's existence, and therefore cannot comprehend its death. It's just simply too foreign to understand. It doesn't help that ORT has a completely different concept of death from anything on earth (as in, it can't be killed by the earth's standards. Even if you manage to destroy the body, that still wouldn't constitute as death for ORT). Although the term "Gaian concept of death" is frequently thrown around, this is never actually mentioned in any canon source, and it's actually something fanon came up with to explain why he can't kill certain beings. Theoretically, there shouldn't be anything to stop him from killing a rock from the planet Mercury or something.
As a side note, the eyes are always active, but as long as Shiki has his glasses on, he can't see the lines. While normally, his eyes are black, when he consciously uses his eyes, they turn a bright shade of blue.