Birds do it, and bees do it--and although we don't know how, how often, or for how long, dinosaurs had sex, too. The reason dinosaur mating is such an enduring mystery is that it's hard to picture a five-ton Tyrannosaurus Rex male putting the moves on an even bigger female, or a pair of Triceratops not goring themselves on each others' horns as they attempt to perpetuate the species. Add the fact that male and female genitals don't tend to persist in the fossil record, and the average paleontologist knows less about dinosaur sex than a second-grader knows about the human variety.
To show how huge a mystery dinosaur sex remains, it's only in the past few years that scientists have been able to distinguish between male and female dinosaurs of the same genus--and even these interpretations aren't accepted by the entire scientific community. Logically, there's every reason to believe that female dinosaurs had bigger hips than males, since females had to carry and lay eggs. Also, there's good evidence that the frills of male ceratopsians were bigger than those of females--large frills being a sexually selected characteristic that helped males to attract mates.
Dinosaur Sex - Reasoning by Analogy with Modern Mammals
Since there aren't any living specimens available for observation, one way to explore the sex life of dinosaurs is to extrapolate back from the largest land animals alive today--elephants and giraffes. With their long necks and squat trunks, giraffes are shaped a bit like smallish sauropods; the way they have sex is that the male approaches the female from behind, keeps his neck low to the ground (so as not to put undue stress on his heart), and does his business very quickly. Elephant males--which vaguely resemble mid-sized hadrosaurs--also approach females from the rear, and they don't linger on the act, either.
The trouble is, reasoning by analogy can only take us so far. As big as it may seem to us, a male giraffe is tiny compared to a full-grown Brachiosaurus; it's hard to imagine even a healthy female successfully bearing its 50-ton weight. And a big reason full-grown elephants can mate at all is that their tails are laughably tiny; imagine the logistics that would be involved with the long, heavy, bulky tail of a Parasaurolophus female.
Dinosaur Sex - Reasoning by Analogy with Modern Reptiles
Ideally, we could infer all we need to know about dinosaur sex by observing bird sex--after all, birds are directly descended from dinosaurs, and at least some species have presumably retained the sexual habits of their ancestors. But once again, there's a big "uh-oh" here: the biggest birds are orders of magnitude smaller than the biggest dinosaurs, so guessing how stegosaurs had sex by observing the mating habits of chickens doesn't make much sense (though one can make a better case for the roughly chicken-sized Velociraptor).
In this case, we're better off considering the mating habits of another close dinosaur relative: the crocodiles, which branched off from the precursors of dinosaurs, the archosaurs, at the end of the Permian period. Large crocodiles and alligators mate in the water; the male hovers over the female for a few seconds and deposits his sperm. The advantage here is that the natural buoyancy of water reduces the effective weight of the male, so it's tempting to imagine a male and female Apatosaurus venturing briefly into a nearby lake to accomplish the deed. Sadly, though, there's absolutely no proof that this was the case.
Did Dinosaurs Have Sex Organs?
As mentioned above, sex organs--since they're made of easily biodegradable "soft" tissues--are virtually never preserved in the fossil record. For this reason, paleontologists can't say for sure if male dinosaurs had penises, or if female dinosaurs had something even remotely approximating the modern mammalian vagina. (You can stop laughing now: from a biological standpoint, there's no reason a hundred-foot long sauropod would need a penis the size of a Lincoln Town Car, though you have to admit it is an arresting image).
Based on the anatomy of modern reptiles, though, it's more likely that male and female dinosaurs had cloacas--primitive orifices used for urination, defecation, and copulation. Once a male and female dinosaur arranged themselves into the right position, sex by cloaca would have been a simple affair; all that would have been needed was a few seconds for the male to deposit his sperm in close proximity to the female's eggs. Talk about anticlimactic: it's possible that sex between two huge dinosaurs lasted only as long as a good sneeze!