As you would expect from countries who share so much history and are in such close proximity with eachother, the languages of the Scandinavian countries are all very similair and of course stem off the same Northern Germanic section of the language tree.
While Swedish, Norwegian and Danish are all mutually intelligible (relatively), Icelandic and Faroese are much less so, despite also being North Germanic languages, they maintain much closer relations to Old Norse (the 'original' Nord language). Finnish, of course, is completely different to all of these, as a Finno-ugric language it is much more similair to Estonian or Hungarian.
Norwegian today has been heavily influenced by both Danish and German, from the Denmark-Norway times of the 1500s (all the way up until the early 1800s) when Danish was the written language used and taught in Norway, and from the presence of the Hanseatic league in Bergen.
Written Norwegian today still very closely resembles Danish, although spoken is completely different, the accent with which Danish is spoken is often the point of many jokes.
As touched on previously, Norwegian has two written forms and many spoken dialects which differ enormously to written Norwegian. Bokmål is the more common form of written Norwegian and is the version which developed during their times under Denmarks rule, and therefore written Norwegian is extremely similair to written Danish, while Nynorsk (spoken by about 10-15% of the population, at a probable over-estimation) was the newly independent Norwegian governments attempt at revitalizing the language and separating itself a little from Denmark. Interestingly, most spoken dialects are probably closer to Nynorsk than Bokmål.
Anyway, have a look at a basic phrase from the three languages:
Swedish - två svarta katter
Norwegian - to svarte katter
Danish - to sorte katte
The word for 'cats' is almost the same across the board, while the word for 'two' also shows similarities, although Swedish shows a larger difference here to Danish and Norwegian. You can see that the word for 'black' is a little different for each one but still very similair. This is only one example to show how it would be relatively easy to understand eachother despite being three separate languages. I suppose the main difference is in inflections and sentence structure - for example in Norwegian you would be more likely to say (correctly) "the car mine" while in Swedish you would say simply "mine car":
- Bilen Min (Norwegian)
- Min Bil (Swedish)
Anyway, this entry is somewhat boring and a little all over the place, I just felt like mentioning that there are similarities and differences between the languages.
It was proposed once that there should just be one language for all the countries, and in Iceland when they teach Danish as a language they just call it 'Scandinavian' - do you think there should be a merger language for simplicity sake?