Germany consists of sixteen states. While we have a 'Basic Law' that is valid for all states, every state has a certain level of souvereignity, and holidays are part of that, the only exemption being our national holiday on October 3rd. Apart from that, there are eight holidays that all states share, and a few extra one in each states depending on local culture and religion (some areas of Germany a predominantly Catholic, some Protestant). That's why there are some 'holidays' that, while celebrated, are not official holidays according to the 'no work' thing. I myself live in Baden Würthemberg in the very south west of Germany, a predominantly Catholic region (but I'm not), so what I write here doesn't necessarly apply for all parts of Germany.
The first day of the year is also the first holiday for the whole of Germany. I think this is really nice, as everyone gets to sleep in on New Year's Day, after partying the night before :D A little historical excursion; the Romans where the first to set New Year's Day on the 1st of January in 156 B.C., before, it was the 1st of March. Pope Innozenz XII. made it uniform for all of Europe in 1691, although most of Western Europa has used the date since the middle ages. For some time, the church had used Christmas Day, the 25th of December, as New Year, and there was some confusion going on to which year the days between Dec 26 and Dec 31 belonged - that's why they are often still refered to as 'between the years'.
In Germany, as in probably most other countries, New Year's Day starts right at 0:00 with a big bang - fireworks. It's also common to clink glasses with sparkling wine, champagne if you can afford it. Once you return inside from the fireworks, at least we do that in my family, you sit together for a bit longer, drinking tea and eating a New Year's Pretzel (research tells me that the shape depens on the area you're in, for us it's a pretzel). It's made from yeast dough, and heavyly decorated with braids and flowers and sometimes letters as well. The size varies, depending on how many people you have over- there are small, plain ones you can eat alone, and big ones that will feed a family of ten :D It can be a little dry, so I usually eat it with a bit of butter.
On New Year's day itself, it's common to take a walk and visit relatives to wish them a happy new year or 'Prost Neujahr'. Children will sometimes get a bit of money from those visits as well; though I do believe that's mostly bribery to make them come at all ;) While my family didn't practice this, traditional food for New Year's Day is, depending on the region, either lentil soup or sauerkraut - both are said to bring good fortune to your finances for the year!