Shinigami Eyes are eyes which can see both the names and lifespans of Humans floating above their heads.
Obtaining Shinigami EyesEdit
If a Shinigami drops its Death Note in the human world and a human picks it up, that Shinigami can give the human Shinigami Eyes at the expense of taking half of that human's remaining lifespan. The eye deal is first mentioned by Ryuk when he offers Light the trade, which Light refuses, as he is unwilling to give up half his lifespan. It is of course of no consequence to the Shinigami as they can gain more years of life easily through their Death Note.
In the manga, a humans Shinigami Eyes are depicted as having gold irises, with red pupils and pale yellow sclera. In the anime, the irises are simply a bright and shining red; in the films, they are depicted with a yellow "cat eye" appearance. It should be understood that this is merely the representation of a person having Shinigami Eyes for the reader/viewer's benefit, as it is impossible for humans to tell (scientifically or otherwise) who possesses Shinigami Eyes.
Abilities Gained Through the EyesEdit
Humans who follow through with the "eye deal" gain the ability to see other humans' names and lifespan, making it easier to use their Death Note. Although, just as Shinigami cannot see each other's lifespans, humans cannot see their own lifespan or the remaining lifespans of other Death Note users. They also gain 3.6 times their current vision, as revealed by Tsugumi Ohba in Death Note: How To Read 13.
Misa Amane briefly explains to Light that in order for the eyes to work, the person being looked at must be visually recognizable. The eyes will not work if the person's face is obscured or out of view, even partially. Also, apparel (such as glasses) have no effect and do not prevent the name from being shown.
A person's name can be seen in a photograph or even video footage, yet it is impossible to tell a person's name from a drawing (no matter how photorealistic it appears). If the person is deceased, their name will not appear in a photograph or in video footage of them.
In Other MediaEdit
In Death Note: Another Note, the antagonist Beyond Birthday (a former Wammy's student and would-be successor to L) is said to have been born with Shinigami Eyes, enabling him to see the names and lifespans of those around him. It is never explained how a human could be born with the eyes, though the idea of a Shinigami losing his or her eyeball and then dropping it to Earth was mentioned in the story.
Characters Who Possess "Shinigami Eyes"Edit
Misa Amane Edit
In all incarnations of the franchise, Misa Amane is the first character to be depicted as having made the trade. After she is arrested on suspicion of being the Second Kira, Rem erases her memories of the Death Note, refusing to kill her, only wanting the best for her. With her memories of the Death Note gone, Misa loses her Shinigami Eyes in the process. After regaining her memories and uncovering the buried Death Note that Light left in the woods, she makes a second eye deal with Ryuk, quartering her life span.
After the timeskip, Misa continues to use the power of the eyes up until Light orders her to forfeit her Death Note and pass it on to Teru Mikami, causing her to lose her memories and the Shinigami Eyes once more. It can be assumed that after Rem's death, Misa regained a portion of her lost lifespan, though the question of how many years she had left remaining is unknown, as Death Note: How To Read 13 reveals that Misa committed suicide a year after Light's death (this is also implied in the anime).
In the films however, she does not commit suicide and only loses her memories of the Death Note (and the eyes) after Light's death and L burns the notebooks. She is shown a year later, wondering why there seems to be a gap in her life and memories. L does hint in the novelization of L: Change the WorLd that Misa's lifespan was still very much shorter and that she did not have much time to live.
Kyosuke Higuchi Edit
In the manga and anime, Higuchi (as the Third Kira) is at first reluctant to obtain the Shinigami Eyes since, like Light Yagami, he intends to enjoy a full life and the fruits of his work as Kira. He even plots to seduce and marry Misa Amane since her possession of the Eyes could be useful in his plans. Circumstances, however, force him into making the deal with Rem when someone, whose name he cannot otherwise obtain, threatens to name him on national television. Higuchi actually uses the eyes in order to kill a traffic cop who has pulled him over for speeding. In the end though, his time with the eyes is short-lived as he winds up captured and dies shortly after, Light having written his name down on a piece of the Death Note.
Kal Snydar (a.k.a. Jack Neylon) Edit
In the manga, after Sidoh discovers the Mafia hideout, a subordinate of Mello known as Jack Neylon (real name Kal Snydar) is made to go through with the eye deal. In the first failed raid on the hideout, Jack reads out the names of the names of the Special Ops team as Sidoh unmasks each of the members. Each person's name was then written down by Rod Ross in the Death Note. Neylon/Snydar later dies during the second raid, as Light had written his name down in the Death Note, along with other Mafia members. In the anime, the eye deal and the first raid of Mello's hideout were left out entirely.
In both the manga and the anime, Light's father ends up being manipulated into trading for the eyes as part of Light's plan to infiltrate Mello's hideout the second time. Interestingly, Matsuda was the original intended target, but Soichiro accepts instead. With the power of the eyes, he is able to see Mello's true name (Mihael Keehl) but is shot by one of the few remaining Mafia members and is fatally wounded when Mello detonates charges and destroys his hideout. Light desperately tries to persuade his dying father into writing down Mello's name in the Death Note, but Soichiro succumbs to his wounds. He then dies, relieved by the belief that his son isn't Kira, as Light's lifespan was visible at the time.
In the films, Soichiro never gets the eyes and doesn't die, as there is no timeskip or the events of "part 2". Unlike his manga/anime counterpart, however, he does learn that his son is Kira.
After gaining Misa's Death Note, Mikami is revealed to have made the deal with Ryuk. He remains with "the eyes" all the way up to his death. Interestingly enough, although the eye-deal should have cured him of his vision impairment, he still wears his glasses from time to time.
C-Kira ("Cheap Kira"), true identity unknown, is the catalyst of the post-series one-shot, where he/she kills off the elderly and sick, as well as people who wish to die. Though his/her identity is never revealed, he/she is depicted with Shinigami Eyes, presumably having made a lifespan trade with Midora . C-Kira commits suicide at the end of the one-shot manga when Near calls him an "abominable murderer" after announcing that he has no interest in the case. Near, like L before him, will only accept cases that he takes a personal interest in.
Beyond Birthday was born with Shinigami Eyes. He is the killer in the BB murder cases discussed in Death Note: Another Note. Although it is never explained how a person can be born with the Shinigami Eyes, Mello jokes in the novel that a Shinigami might have dropped his or her eyeballs to earth instead of a Death Note.
Kiyomi Takada (film) Edit
In the second film Death Note: The Last Name, the Yotsuba story arc is removed, with Sakura TV and Kiyomi Takada serving in place of The Yotsuba Group and Higuchi, respectively. Takada gains the Shinigami Eyes from Rem, but as in the case of Higuchi, winds up captured as Kira and killed by Light's piece of Death Note.