So, you're worldbuilding a scifi alien species or a fantasy race that can reproduce as both male and female. Or you're a biologist or science educator. And you want what you're writing/making to be inclusive to intersex people.
The term hermaphrodite in biology refers to species that can reproduce as both male or female. The problem is it's also a slur when used against intersex people (it's also incorrect - we're not hermaphrodites). I recently polled other intersex folks informally and a plurality were fine with the h-word being used with the correct meaning in appropriate contexts. But the qualitative feedback I got was, all things equal, most people who wrote to me said they'd just be happier seeing the term less.
So here is a glossary of words you can use to talk about the biology of sex without invoking the h-word:
Gonochoric species where individuals (typically) reproduce only as either male or female. This means the species produces two types of gametes that have different sizes, and conventionally the larger is female. Gono- for generation/reproduction and -choric for separated/distinct. Note that "non-gonochoric" includes both hermaphroditic species and those that can reproduce asexually (e.g. parthenogenesis).
Gonosyne: species where individuals (typically) can reproduce as both male or female. Gono- for generation/reproduction and syne for together/combined. This is a term I have coined to be an alternative to hermaphrodite.
Types of gonosyny: it's common to categorize different forms of gonosyny based on temporality and how many gonads an individual has.
Grouping by Temporality:
Cosex: species where individuals can reproduce as both male and female simultaneously. Alternative terms: cosexual, simultaneous hermaphrodism. For example: land snails/slugs typically mate by linking up both pairs of genitals.
Dichosex: gonosynic species where individuals reproduce as male and female at different times in their lives. Protandrous species start as male then switch to female; protogynous start female then switch to male. Some species cycle between the two (serial/bidirectional hermaphrodism/disexuality). Alternative terms: dichosexual, dichogamous, sequential hermaphrodism. I coined this one after feedback that "dichogamous" was not intuitive to non-botanists, keeping dicho- (in two parts/paired) for simplicity. Example animal: clownfish (the Finding Nemo fish).
Grouping by Gonads:
Digonic: species that can reproduce as both male and female because they have separate male and female gonads. Digony can be cosex or dichosex. In botany the term monoecious is used for flowering plants. For example: barnacles have their ovaries in the base of their body, and testes in the back of their head.
Syngonic: species that can reproduce as both male and female, because their gonads can produce both male and female gametes. Alternative terms: syncoecious, monoclinous, ambisexual. (Note ambisexual has other meanings.) Syngony can be cosex or dichosex. In dichosexual species the gonad changes which gametes it produces when the individual changes sex. For example: land slugs have a single gonad (ovotestis).
Together this makes four categories. Examples come from: Sex change in plants and animals: a unified perspective.
Syngonic cosexual. Simultaneous gonosyny within the same flower/gonad. Examples: Black-jack daisy, Lady of the Night cactus; mangrove killifish, stubby-root nematode.
Syngonic dichosexual. Non-simultaneous gonosyny within the same flower/gonad. Examples: Bromelia chrysantha, grape ivy; California sheephead fish, common limpet.
Digonic cosexual. Simultaneous gonosyny from different flowers/gonads. Examples: bitter melon, jaraguá grass; barnacles, flatworms.
Digonic dichosexual. Non-simultaneous gonosyny from different flowers/gonads. Examples: papaya, catsfoot; staghorn coral, earthworm.
Hope this is helpful! I have two little notes I want to add on: Note on "non-gonochoric": it's possible this could also include isogamous speries? There's some ambiguity in use. Isogamy refers to sexual reproduction where you don't have two different sized gametes - instead it's two identically sized gametes that are getting combined. This is the standard amongst unicellular eukaryotes and very common in fungi.
Note on ambisexual: this is the term that Ursula K LeGuin used for the dichosexual aliens in the Left Hand of Darkness. The term in biology these days refers to undifferentiated (immature) tissue that has yet to develop into a given sex - e.g. a human embryo has ambisexual gonads until sexual differentiation later on in development. The term ambisexual has a ton of other meanings in other contexts such as a sexual orientation. For this reason I'd personally avoid it.