New Years Eve is one of the best nights in Iceland, in my opinion. My first New Years, we were stood outside Hallgrimskirkja (the big church on most of the Reykjavík postcards) and the atmosphere was amazing, with everyone so enthusiastic and feeling the community. It seemed half the city had showed up. When the fireworks started properly, it was even more unbelievable.
It can't be called a firework display as such, as there was no coordination or anything like that, but it was best show of them I've ever seen. I have never ever seen that many going off at once, at any event or time anywhere in the world. From the church, you can see them from every angle. The photographs you can find of what's going on are no exaggeration. In fact, I'm not sure I've seen one that truly demonstrates the scale of the event.
Of course, so many crowded people setting off so many fireworks at night has its dangers, but there are goggles on sale around the time you can protect yourself with - major injuries from the fireworks come when something hits an eye. Even if you don't want to stand in goggles on New Years Eve, if you are travelling with children, it might be wise for them!
The fireworks are on sale all over, but the Rescue Team is one of the biggest sellers. As a volunteer organisation that every day saves visitors and locals from sticky situations, its one of their most important times for raising funds, so I recommend buying off them.
After the fireworks, the party tends to stay on all night, like most other places. Prepare for some crowded bars and steep drink prices, but an amazing atmosphere and a wealth of fun people. Its also a great night to meet locals rather than just largely tourists, as most Icelanders want to make the most of the turn of the year too.
Hope that helps and you have an amazing time over here!