That's actually the Battle of the Black Gate, for the record.
At any rate, the LOTR wiki states that, in the book, there we're indeed horseback cavalry, about 1000 of them. They staved off Harad archers, while foot soldiers from Gondor battled the orcs and trolls.
From the wiki;
Against the Host of the West was arrayed all of Sauron's hordes of orcs, Trolls, and barbarian Mannish allies such as the Easterlings and Southrons (Haradrim). An exact count is not given of the number of Sauron's forces, but it is said that they were "ten times and more than ten times" greater than the Host of the West, making it a force of at least 60,000. Sauron's forces surrounded the Armies of the West on three sides, with the Easterlings on the left flank, the Orcs held the center, and the Haradrim on the right. The Armies of the West always had cohesive stability within their center and kept the Enemy from breaking through by force of numbers. The solid infantry squares of Gondor infantry beat off their inferior Orcish opponents while the Rohirrim cavalrymen staved off the archers of the Harad. The small force of Easterlings launched a quick attack before being repulsed by the armies of Gondor and Rohirrim while the larger Haradrim force slowly retreated to the hills. By now, pressure had eased off of the flanks of the Armies of the West, who smashed into the lines of the Orc-host, slowly beating them off and forcing a retreat back through the Black Gate (though not without suffering sizable losses of their own).
In the movie, it really isn't explained why they aren't present in the battle. From reading other forums, most people say that Aragorn (and the others) sent the horses to a side, to protect them from combat. Fighting from horseback is a difficult feat, and, most of the time, cavalry forces in the medieval ages would only engage the enemy for a few seconds before pulling back, to avoid being pulled off their horse by enemy infantry, having the horse fall on the rider in the event that the horse is stabbed by a spear, etc.