When tourists ask me what they should see/do if they are only in Stockholm for one day, I usually say that they shouldn't miss The Vasa Museum and, especially if it is good weather, Skansen.
Skansen is a combination park, outdoor cultural museum and zoo. It is located on a hill on the island of Djurgården (where many other attractions are located). Founded in 1891, Skansen is the world's oldest open-air museum and one of the most popular attractions in Stockholm. The founder, Artur Hazelius, had around 150 buildings (from churches to manors to farmhouses) from different parts of Sweden shipped to the museum piece by piece... so you get a real feel for traditional Sweden. The buildings are all open to the public and there are usually people in traditional dress there to explain a little about the building and the traditions.
Skansen is also the perfect place to see how Swedish traditions and holidays are celebrated (Easter, Christmas, Midsummer, etc;). They have great Christmas and Easter markets every year. I was just there on June 6th which is the Swedish national holiday... park-goers were there waving their flags and having picnics. The king and queen were also there in the evening, front-row at a traditional concert.
The zoo is spread throughout the park with everything from exotic farm animals to Scandinavian wild animals like wolves, seals, bears, moose and reindeer. It really is fun for the whole family!
Skansen is open daily, year-round and entrance costs 120 SEK (50 SEK for children and 100 SEK for seniors). Free entrance with the Stockholm Card.
To get there: Bus 47 or a street car/tram from Normalmstorg or else the Djurgård ferry from Slussen or Nybroviken.