Hermaphrodite species are those in which the separation of sexes exist (there are female and male gametes), but only in a genetic level, as all individuals produce both male gametes (sperm cells) and female gametes (egg cells).
It is vital to note the distinction between hermaphrodite species and asexual species, since the two can get confused often. In short, asexual individuals are neither male nor female, while hermaphrodite individuals are both male and female, usually but not necessarily at the same time (the alternative is sequential hermaphroditism, in which all individuals start life as one sex and change to the other spontaneously as they age). In practice: while asexual species reproduce individually and never mate (though they might mate "symbolically" and not exchange genetic material as is the case of the Asari for instance); hermaphrodite species usually do mate, although in rare cases they might not need to, being able to impregnate themselves. Some hermaphrodite species actually fight over which of the pair will become pregnant once they mate (this is the case of some real life flatworms).
Note 1: The term "androgynous" is often used in fiction, and refers to the broader notion of having both male and female features in the same individual. In the biological sense, calling a species androgynous (as opposed to an individual) will probably mean that they are hermaphrodites.
Note 2: There is also a condition called dioecious isogamy, in which the species is dioecious (separate sexes) but the sexes do not correspond to our notion of "male" and "female" because the two kinds of gametes are indistinguishable. For a casual observation, it's likely that there's no easy way to distinguish between hermaphrodite species and dioecious isogamous ones, as both of them reproduce sexually without having male and female individuals. To find out which scenario it is would require looking at their gametes' morphology, and even then it would be extremely hard to distinguish between hermaphrodite isogamy and dioecious isogamy. To help avoid these traps, only add here species which are explicitly identified as hermaphrodites.