Sexual dimorphism-a pronounced difference in size and appearance between the adult males and the adult females of a given species, over and apart from their genitalia-is a common feature of the animal kingdom, and dinosaurs were no exception. It's not unusual for the females of some species of birds (which evolved from dinosaurs) to be larger and more colorful than the males, for instance, and we're all familiar with the giant, single claws of male fiddler crabs, which they use to attract mates.
When it comes to sexual dimorphism in dinosaurs, though, the direct evidence is much more uncertain. To begin with, the relative scarcity of dinosaur fossils-even the best-known genera are usually represented by only a few dozen skeletons-makes it perilous to draw any conclusions about the relative sizes of males and females. And second, bones alone may not have much to tell us about a dinosaur's secondary sexual characteristics (some of which consisted of difficult-to-preserve soft tissue), much less the actual sex of the individual in question.
Female Dinosaurs Had Bigger Hips
Thanks to the inflexible requirements of biology, there is one surefire way to distinguish male and female dinosaurs: the size of an individual's hips. The females of large dinosaurs like Tyrannosaurus Rex and Deinocheirus laid relatively large eggs, so their hips would have been configured in a way to allow for easy passage (in an analogous way, the hips of adult human females are noticeably wider than those of males, to allow for easier childbirth). The only trouble here is that we have very few specific examples of this type of sexual dimorphism; it's a rule dictated primarily by logic!
Oddly, T. Rex appears to have been sexually dimorphic in another way: many paleontologists now believe that the females of this species were significantly larger than the males, over and above the size of their hips. What this implies, in evolutionary terms, is that female T. Rex were particularly choosy about selecting mates, and may have done most of the hunting as well. This contrasts with modern mammals like the walrus, in which the (much bigger) males compete for the right to mate with smaller females, but it's perfectly in sync with (say) the behavior of modern African lions.
Male Dinosaurs Had Bigger Crests and Frills
T. Rex is one of the few dinosaurs whose females asked (figuratively, of course), "Do my hips look big?" But lacking clear fossil evidence about relative hip size, paleontologists have no choice but to rely on secondary sexual characteristics. Protoceratops is a good case study in the difficulty of inferring sexual dimorphism in long-extinct dinosaurs: some paleontologists believe that the males possessed larger, more elaborate frills, which were partially intended as mating displays (fortunately, there's no shortage of Protoceratops fossils, meaning there are a large number of individuals to compare). The same appears to be true, to a greater or lesser extent, of other ceratopsian genera.
Lately, much of the action in dinosaur gender studies has centered on hadrosaurs, the duck-billed dinosaurs that were thick on the ground in North America and Eurasia during the late Cretaceous period, many genera of which (like Parasaurolophus and Lambeosaurus) were characterized by their large, ornate head crests. As a general rule, male hadrosaurs seem to have differed in overall size and ornamentation from female hadrosaurs, though of course, the extent to which this is true (if it's true at all) varies significantly on a genus-by-genus basis.
Feathered Dinosaurs Were Sexually Dimorphic
As mentioned above, some of the most pronounced sexual dimorphism in the animal kingdom is found in birds, which (almost certainly) descended from the feathered dinosaurs of the later Mesozoic Era. The trouble with extrapolating these differences back 100 million years is that it can be a major challenge to reconstruct the size, color, and orientation of dinosaur feathers, though paleontologists have achieved some notable successes (establishing the color of ancient specimens of Archaeopteryx and Anchiornis, for example, by examining fossilized pigment cells).
Given the evolutionary kinship between dinosaurs and birds, though, it would not be a major surprise if, say, male Velociraptors were more brightly colored than females, or if a female "bird mimic" dinosaurs sported some kind of feathery display meant to entice males. We do have some tantalizing hints that male Oviraptors were responsible for the bulk of parental care, brooding eggs after they were laid by the female; if this is true, then it seems logical that the sexes of feathered dinosaurs differed in their arrangement and appearance.
A Dinosaur's Gender Can Be Hard to Determine
As stated above, one major problem with establishing sexual dimorphism in dinosaurs is the lack of a representative population. Ornithologists can easily collect evidence about extant bird species, but a paleontologist is lucky if his dinosaur of choice is represented by more than a handful of fossils. Lacking this statistical evidence, it's always possible that the variations noted in dinosaur fossils have nothing to do with sex: perhaps two differently sized skeletons belonged to males from widely separated regions, or of different ages, or perhaps dinosaurs simply varied individually the way humans do. In any case, the onus is on paleontologists to provide conclusive evidence of sexual differences among dinosaurs; otherwise, we're all just fumbling in the dark.