December is a special time to be in Florence - the winter light can make for fantastic sunsets, and the Arno can be very still at times, providing some great picture taking opportunities. There are Christmas markets everywhere, and if you are really lucky you may even see some snow - while it is rare to actually have snow fall in the city center (you will see the hills and mountains in the distance are covered) yet there were back to back storms on the same night in December in 2009 and 2010.
My favorite things about Flornece in December though are the different styles of Christmas lights hanging all over the city center, and the general festive feeling in the cold winter air.
A great thing to do with kids in Florence this time of year is ice skating at the Parterre near Piazza Libertà (Pattinaggio sul ghiaccio in Italian) - you can rent the skates so don't worry about brining your own.
And of course there are Christmas (Natale) markets everywhere - many are just one specific weekend. One of our favories is the "Nataleperfile" at the Palazzo Corsini (offical site). Not only is this a great place to find gifts, but they usually have some pretty amazing choices for lunch, and you get to see the palazzo up close. If you are looking for a panettone, this is the place.
There is also the German Christmas Market in Piazza Santa Croce - this is a big market mostly featuring handicrafts and food from Germany and Austria. It can be a lot of fun to just hang out and drink some mulled wine and perhaps try a bratwurst if you are not in the mood for shoppping, but there are plenty of gift ideas here too, including cheeses, salamis, and sweets.
Another market to check out for some great gifts and food is the Organic Market in Piazza Santo Spirito - they usually have a date during a weekend in December, and sometimes an extra weekend in Piazza Santissima Annunziata also.