This article or section contains major spoilers for the plot of Blood is Mine Chapter 2: Blood on Mars and onward. New readers are advised to proceed with caution.
The red as a concept has been described by Morgenstern as "a sort of infinite meat dimension" that Jane, Jane's mother, and others like her could draw material from to grow their bodies or "externalize" constructs (such as Mom's Red). The blood affords them a link to the red, and they need only draw from it and give what they externalize shape. Jane has pulled material from the red only once, to repair her damaged shoulder in the warehouse. It did not go well. Due to her inexperience, she very nearly allowed what she was externalizing to grow out of control, and even when she was assisted in this by the calming effects of Level, she still ended up putting a mess of extra bones and teeth where they didn't belong.
In order for that which bears a red to externalize any material, it must grow directly from a host's body. Jane cannot externalize material from the red through a pool of her blood, for example. Any constructs she desires to externalize must grow from her. Morgenstern has described it as growing a tumour, and giving it sentience, before severing it from one's body. This has been discouraged by Red, as Jane is not entirely skilled at putting her body back the way it's meant to be.
Aliens that bear a red, according to Mom's Red, have no innate fondness for one another. If one of their kind becomes aware of another in existence in the cosmos, it will not rest until that other is destroyed. Jane lacks this innate fury, likely because of her half-human nature.
Red is a psychological manifestation of Jane's connection to the red as described above. She is essentially the non-human parts of Jane's mind given form. She is the half of Jane that wants her to "stretch thin and drink deep of the stars", according to Mom's Red. Originally, when Jane was still discovering her abilities, her red went to great lengths to try to scare her, and appeared to her mentally in the form of a giant mouth, and also as a stretched out version of Jane. Since Jane has conversed with her more, and the two halves have reached an level of understanding of each other, she has take the form of Jane in a bloody nurse uniform with blood red hair, leading to her being given the name "Red". She can manifest to Jane (and Fuse, after an extended period of syncing) in the real world, but she does not actually exist, nor can she interact with her environment. She can also be contacted inside Jane's mind, where she is shown standing in a slightly altered version of Jane's childhood bedroom.
Since Jane has been speaking to her more, Red has been shown to be a source of a great deal of information regarding Jane's non-human side. She has also been shown to care about Jane's friends (she is very bothered by the idea that they could die) and gets very, very angry when people mess with those friends.
Fuse was shown to be able to contact her as well in the school, however this may just be him attempting to visualize talking to himself, and subconsciously figuring that Red is what that would probably look like. It is unlikely Fuse has a red, as he is fully human and bears no connection to it.