Iceland is so far north, if it isn't being confused with Greenland, people often forget it's existance! It is a volcanic island, and although it is located just south of the Arctic Circle, it's kept warm by the gulf stream and all the thermal energy boiling below its surfaces (most of the energy used in Iceland to heat houses, hot water, etc. comes from the natural thermal energy). Even though it is warmer for being so far north, the weather can still be very cold, and more than 10% of it is covered by glaciers.
The capital city is Reykjavik, and it is the northernmost capital city in the world, although it's located in the southwest part of Iceland. It is also where the majority of people live. If you are flying into Iceland, there is an international airport in Keflavik, which is about 40km from Reykjavik, and is located the southwest, even more southwest than the capital. Be aware that like most things, taxis will be expensive. It would probably be cheaper to try to take a bus if possible, or likely free if you try you hand at hitchhiking, if weather permits. Iceland is among the safest countries in the world. If you need to get around the island, by plane is an option, if it's in your price range.
If you like horses, you should consider trying out the unique Icelandic horse - a small and sturdy breed full of personality. You can go on anything from a half-day to over a week-long ride, and it's a wonderful way to experience the countryside. There are two major horse riding centres near Reykjavik, Ishestar and Eldhestar. Both make arrangements with the bus tour companies, or you can contact the centres directly.
Iceland is home to Puffins (the birds with the colorful beaks that also live in Alaska), dolphins, many various kinds of birds, and the Icelandic sheep (it only exists in Iceland). It's a great country for nature-loving, active people.