Elves have, so far as we can tell, the same sexual organs as everybody else. Their clothes certainly cover up the same areas of the bodies. (This is quite odd, when you come to think about it, because sexual modesty implies a loss of innocence, and elves are free from original sin. But their souls and their bodies -- their fëa and their hröa -- are wired differently from those of mortal men, doomed to die.) According to custom, female elves are more likely to be healers, and male elves are more likely to be warriors, but there is no role that a female elf is prohibited from performing just because she's female. Nevertheless, there is an essential difference between the genders: if a male elf dies he always reincarnates in male form, and if a female elf dies, she always reincarnates in female form. (This, incidentally, is true of the
gods Valar as well: although they are incorporeal and wear bodies when they have dealings with humans, some always wear male bodies and some always wear female bodies.)
The elves reproduce in the same way as all other mortals, although it is hard to imagine a hobbit doing it, isn't it? Elf marriages are like English "common law" marriages: the act of sexual intercourse is sufficient to make two elves married. In practice, the elves do perform solemn ceremonies of marriage and betrothal but it isn't the ceremonies which make the marriage. (By tradition, the brides mother gives the bridegroom a necklace to signify betrothal, a point which was presumably not lost on Aragorn.)
Elves marry for life. Since dead elves are reincarnated, it is pretty bad form for a widow or widower to remarry. Finwe did marry Indis after Miriel died, but that was a source of ill-feeling between Feanor and his half-brothers Fingolfin and Finarfin.
After they have had a few children, an elvish couple lose interest in sex, and dedicate themselves instead to other elvish pursuits: sitting in idyllic woods idyllic composing harp music about idyllically sitting in woods idyllically composing harp music; idyllically baking lembas; idyllically fighting genocidal wars about the ownership of magic gems. After their children have grown up, which takes millennia, they may actually live apart. There is really no such thing as elvish lust: affection, sexual desire and the bearing children go together, almost, one might say, like a horse and carriage. You really can't have one without the other.
Daily Mail readers believe that this is also how human sexuality works. But it isn't.