What is Otherness?
On a basic level, "otherness" to me is just the quality of being other, not part of oneself, not part of a group one is a member of, or simply not that -- or not part of that -- which is the subject of attention. It is what is foreign, alien, alterior, alternative to a particular X... it's not-X.
On a deeper level, however, the necessary act of definition-by-exclusion as "other" creates a relationship to an agent, constitutes an action by an agent on that "other", such that "otherness" entails having been actively distinguished and segregated out (from self, group or subject) on the basis of one or more points of difference (perhaps arbitrary, perhaps common to "other others"). Hence the verb-form of "othering" as an act of discrimination. The "other" here is not just alien but alienated, excluded from X, quite possibly with the aim of consolidating X as a stable system.
Digging deeper still, emergent consistencies between arbitrary acts of definition-by-exclusion serve to establish boundaries of normativity (of self, group, subject) by which less common but nevertheless natural things (types of behaviour, people, whatever) are deemed essentially non-normative -- abnormal, unusual. As these boundaries become nomological systems the "other" acquires a quality -- in psychological terms -- of being considered essentially unnatural. Beings or events which infringe (or which we suspect of infringing) the laws of normality acquire a quality of strangeness I'd relate to Todorov's fantastique and Freud's uncanny. "Otherness" in that respect may be a projected quality of transgression.
Since ultimately all things we can practically identify as "other" must be as much part of the big system of the cosmos as us (else, we can't practically identify them), the act of definition-by-exclusion is, I think, comparable to Kristeva's idea of abjection. It can be seen as a rejection of that which is or was at one time, in some deeper sense, part of us. (Hence the "familiar" part of Freud's foreign-yet-familiar uncanny. Hence the figuration of the "other" in the form of doppelganger or shadow.)
Ultimately, "otherness" means, for me, the abject. In terms of "a quality of life that sometimes authors attempt to express in fiction", as in "writing the other", it's the experience of abjection as a social process I'd be trying to represent, what it is to suffer the effects of that definition-by-exclusion. In one of its most extreme examples you can see that in the Mandingo trope, the way black men have been rendered abject, subject to a very specific form of discrimination which treats them as symbols, spectres loaded with all the brutality of "base passion" considered expunged by civilisation.
"Otherness" is having to live with that.