The giving of gifts was in the context of the general merry-making of the festival, which began at a sacrifice and banquet (open to the public by the Late Republic) at the Temple of Saturn in the Forum Romanum. From there, revelers dispersed to their homes. There was a tradition that masters wait on their slaves at the gathering within the household. This was probably a very ancient custom, from an era when more Romans lived agrarian lives and therefore worked side by side with their slaves. I suspect this particular tradition was not always observed on the Palatine hill in the Late Republic!
At any rate, there was a family-oriented gathering in the home. Friends could be invited, depending on circumstance (so just like modern western custom of Christmas dinner). Some were probably large affairs, especially among the wealthy, but others were modest. Gift-giving was not the center of the observance, and the range of gifts was apparently not very great.
Children often got sigillaria--little ceramic objects of many shapes and sizes. Some were shaped like little animals (cows or birds, for example); others were oil lamps, which could also be shaped like things; others were flat plaques with a scene inscribed on it. It could be a mythological story (like the very popular Herakles) or something out of the contemporary context (like Pompey's likeness, or a representation of a triumph, etc).
Adults got wax candles (cereī), and this seems to have had little to do with the sorts of motivations for modern gift-giving. The candles were involved in the fertility rites of Saturn (some argue that they were to mark the New Year instead). I have read that the candles, in early days, were thought to be involved in the annual (and ritual?) inspection of subterranean grain storage vaults, but I can't find that mentioned anywhere now (correction: it was Versnel; thank you google books, and thanks to the AH mods for helping me discover google books!). Macrobius says that the sigillaria had originally been meant to be offered to Saturn as proxies for themselves the bodies/lives of the devotees (a common switcheroo in ancient ritual practice), and that the candles were also involved in that ritual. He doesn't really explain how or why they came to be given person to person as gifts, however. As far as this particular topic, I don't put a lot of weight on Macrobius' explanations, though in general I think he's damn good on Roman religion.
Another, and probably common, category of gift was food and drink. A lot of it was probably collective: the master provided some better food and drink during the Saturnalia, some of which may have been offered to the slaves during this customary gathering, if it was observed. It was in general a time to break out the better wines, to eat the better meat, and to enjoy family and friends. Even the grouchy Cato (Agri. 57) says the Saturnalia was a time to dole out some good wine!
For more on Roman festivals in general, see the excellent book by Scullard (Festivals and Ceremonies of the Roman Republic, 1981).