Small spoilers ahead. Anybody who saw Ex Machina knows that director Alex Garland likes his details, and his newest feature, Annihilation, is no different. Every little detail is important when it comes to Lena (Natalie Portman) and her journey into the mysterious phenomenon known as the Shimmer. That includes her ink. Lena's tattoo in Annihilation of a snake wound in the infinity symbol, eating its own tail, appears multiple times throughout the film, but that's not what makes it strange. What makes it odd is how it appears and disappears on multiple characters, including Lena's teammate Anya Thorensen (Gina Rodriguez) and her husband, Kane (Oscar Isaac).
The tattoo in Annihilation, placed on Lena's left forearm, is only glimpsed here and there, and it's an easy detail to miss. So, the first time you spot it, you might think it's just something you overlooked before. The tattoo is big, yes, but it's not showy, depending on the angle it might as well be a birth mark or a cut from Lena's expedition into the Shimmer. Eagle-eyed viewers, however, will note that Lena does not always have this tattoo on her arm. She didn't have it when she went into the Shimmer. Below are two shots from the film: one that takes place before Lena enters the Shimmer and one after. Though it's difficult to see her left forearm, there don't appear to be any markings on Lena's arm in the first photo.
In the second photo, however, the tattoo is definitely visible. (Turn up the brightness to see it clearly.)
It's possible that the tattoo was hidden in the first photo, or that its absence was a trick of the light or a production mistake, but those possibilities wouldn't explain how both her husband and Anya would also have the same tattoo. Lena and Kane were both in the Army, which could give a reason, but Anya was a paramedic with no mentioned military experience. Furthermore, in her early days in the Shimmer, Lena notices a mysterious bruise on her forearm. A bruise that, one can assume, becomes the tattoo.
The tattoo in Annihilation seems to be a mark of the Shimmer. The team determines that the Shimmer is refracting DNA, combining all living things and changing them simultaneously. Maybe someone in the Shimmer had it, and now the tattoo is being refracted onto other people, or it's a mark of the entity itself. When Anya looks down at her hands late in the film, she says she can see her fingerprints moving. Her skin is changing, her DNA altering, all because of the Shimmer. That scene is the first clear shot of Anya's tattoo, which was missing earlier in the film, as seen in the photo below.
The symbol itself, a snake twisted into the infinity symbol eating its own tail, is known as an ourobos, and has had many different meanings over centuries. In medieval times, according to SymbolDictionary.net, the ourobos represented "continuous renewal," life bleeding into death, and a coming together of opposites. The symbol, then, represents what Lena discovers about the Shimmer when she notes that the plant life and animal life in it is "stuck in a continuous mutation." The ourobos is also believed to be linked to early observations of the Milky Way, which means the tattoo could be hinting at the extra terrestrial nature of the Shimmer in Annihilation. Finally, the ourobos signifies self destruction — a snake eating itself — which is another major theme of Annihilation. In one particularly telling scene, Dr. Ventress (Jennifer Jason Leigh) tells Lena, "Almost none of us commit suicide. Almost all of us self destruct."
It's worth noting that Dr. Ventress and Josie Radek (Tessa Thompson) do not appear to have the tattoo, nor do they develop it in the Shimmer, and none of the characters ever mention it. Why some people in the movie have the tattoo and others don't is a question each fan will have to answer for themselves.