The (pre-Christian) Greeks and Romans didn't have a "Christmas" as such, but like most cultures, they did have a quasi-religious get-drunk-and-eat-food midwinter thing.
In the pre-classical Agean, the Lenaia (or Lenaea) was a female winter festival, honouring the 'Wild Women' or Maenads, and Dionysos, the gods of wine, intoxication and theatre. You might remember Maenads from the frankly awful second half of True Blood series 2. According to wikipedia, it originally involved a man or a bull going out onto the hillside to be torn to shreds by these maenads (presumably women acting the part, rather than actual supernatural beings). It is hard to know how much credence to give to these claims; the Greeks tended to have quite outlandish myths to explain their customs (see for example the Athenian founding myth), and to my knowledge there is nothing explicitly written about the origins of this festival. In these situations, scholars tend to try to deduce what is happening from scenes on vases, such as the one below. This is not always completely accurate.
In Athens, they celebrated the Lenaia for the whole month of January (give or take; they didn't have the same monthly calendar as we do). But by the Classical period (5th and 4th centuries), it had become a minor dramatics festival to honour Dionysos with wine and the putting on of plays. Athens held a lot of dramatics festivals, since these were the chance to put on plays. Almost all plays were written and performed for a festival. The Athenians almost never revived plays, so the reception of the audience on first showing was incredibly important. The plays produced were in direct competition with each other.
The Lenaia was originally a comedy festival, but around 432BCE they introduced a tragedy competition as well. Usually around five comedies competed, and four tragedies. Winning such a competition could bring you fame and (indirectly) fortune. A winning playwright could expect to be invited to all the important parties, and be introduced to all the important people on the politics and arts scenes.
The audience probably included Athenian citizens who dwelled in Athens, and citizens who farmed the land around Athens, known as Attica. While Athens was famous for its dramatic festivals, the Lenaia was unlikely to have attracted tourists from too far around, as sea travel was unsafe in winter, as were long journeys overland.
The Saturnalia is one of the better known Roman festivals, as it bears quite a resemblence to Christmas. Privately, it was observed by a series of alterations in Roman 'cultural norms'. Slaves could disrespect their masters, and often ate with, or were served by them. It is hard to know how much they could get away with though; presumably they could still be punished after the festival. There was also the giving of small gifts between family members and friends, and a large public banquet, after which people would often call out "Io Saturnalia!" to each other. This is roughly the same sentiment as "Merry Christmas!". The Saturnalia was also a holiday from all kinds of work: schools and courts were closed, exercise regimes were suspended. While the toga was the usual dress, people instead wore colourful Greek 'dinner clothes', which were usually considered to be bad taste (knitted Christmas jumpers anyone?).
In the Imperial period (AD 17 onwards) there was also a "King of Saturnalia", whose commands had to be obeyed, no matter how ridiculous. Nero supposedly played this role in his youth.
So, this Christmas, why not take a break from your usual festivities, and instead try drinking wine, watching comedy, giving presents and eating too much food?
Io Saturnalia everyone!
Note: I know these were not the only mid-winter Greek and Roman festivals, but they're the two I chose to run with. Don't like it? Do your own blog.