Nordics is made up of five countries (Denmark, Norway, Sweden, Finland and Iceland) and three territories (Faroe Islands, Greenland and Åland Islands).
Scandinavia is used to reference just the three individual kingdoms of Denmark, Norway and Sweden. The histories of these three countries are more tightly intertwined than the rest of the Nordic countries. Scandinavia is known for high taxes, socialized health care and a high standard of living.
Denmark is the gateway to Scandinavia. It was once the home of Viking raiders. For years, Danes have consistently rated themselves as the happiest and most content people in the world.
Norway is the world's 5th-largest oil exporter and the second wealthiest country per GDP in the world.
Sweden was a military power back in the 17th century but has not fought in a war in almost 200 years. Its policy of armed neutrality kept in out of both world wars.
Finland was part of Sweden from the 12th to 19th century and then was a part of the Russian Empire for more than 100 years. While Danish, Norwegian and Swedish are very similar languages, Finnish is completely different. I thought Czech was bad with 7 different grammatical cases but Finnish has 15 different cases.
Iceland was settled by Norwegian and Celtic (Scottish and Irish) immigrants in the 9th and 10th centuries. It was independent for more than 300 years before being ruled by Norway and then Denmark. Located in the mid-Atlantic it is the USA's closest European neighbor.
The Faroe Islands are about halfway between Iceland and Scotland. The islands are a part of the Danish realm but have been an autonomous region since 1948. It is responsible for internal affairs while Denmark is responsible for justice, defense and foreign affairs. The islands are not part of the EU.
The Åland Islands are located in the Baltic Sea. It has been an autonomous province of Finland since 1920.
Greenland is the world's largest island and over 80% is covered in ice. Vikings from Iceland arrived in the 10th century. It was colonized by Denmark in the 18th century. Greenland is still an autonomous part of the Danish realm but Denmark is still responsible for foreign affairs and defense.
Update: Here's a video I found out on YouTube about Scandinavia.