I currently charge my iPhone with a "regular" charger purchased in the U.S. Will this charger work in Europe?
Yes, this will work, although you'll need an adapter plug to be able to plug your charger into a European socket.
Actually, you don't really need your charger as any regular USB port will do. So if you have your laptop with you, you only need your laptop charger (for which you will also need an adapter plug).
A universal travel adapter is something which should be in your travel luggage anyway:
Apple sells a "World Travel Adapter Kit" http://store.apple.com/us/product/MB974ZM/B with a number of different charging options.
It includes the USB-iPhone cable, an iPhone/iPad power adapter (semi-useless but I guess it's always good having a spare at home), and a number of swappable plugs for various countries.
The advantage of this option is that the sliding plugs work both with your USB charger and with any Apple adapters with the sliding plug. That is, you can charge your MacBook, iPad, iPhone, iPod, and old PowerBook all with the same kit.
Because it's basically a USB charger/adapter, you can charge other USB items with it as well, and it's very compact compared to some of the other options.