If you have one day or less

Old Town ("Gamla Stan")

It is easy to get there and easy to get back to your apartment. Just go to the subway stop by the same name, or if you are not in too much of a hurry, go to "T-centralen", exit Sergels Torg, and walk Drottninggatan south until you cross two bridges (that is Parliament) to arrive on the island that is Old Town. It is nice to just walk around, or you could have coffee at Kaffekoppen, beer and burgers at the basement pub Wirströms, or whiskey and Swedish microbrews at Glenfiddich Warehouse.

Vasa Museum

This museum is built around a big warship that sunk in 1628 and was pulled out of the water in the 1960s. The guided tour is usually good, but there is the occasional bad guide, then just leave the tour. The permanent exhibits that are interesting include the "whose fault was it" movie, the "pulling it out of the water" movie (somewhat longer), the small exhibits on life on the boat and maritime warfare.

City Hall

Guided tours only. What physicist would miss the venue of the Nobel dinner? Get there early or stay afterwards to stroll the gardens (not very big) and climb the stairs to the tower for the best view of Stockholm.

For two days or more

Skansen Open Air Museum

If the weather is good this is very nice. They moved traditional houses here from all over Sweden and crafts people do their crafts in them wearing traditional gear. If you stand around they will invite you to participate.

Archipelago Boat Trip

There are around 20,000 islands in the archipelago so why not see a few? Sandhamn is a popular destination but it is a long ride. Fjäderholmarna is closer.

Drottningholm

It's not Versailles but it's a nice palace where the royal family actually lives (unlike the palace in Old Town) and the boat ride is enjoyable on a nice day.

Vikings

People tend to complain that there is no viking museum in Stockholm. That's right, there isn't one, but the National Historical Museum in central Stockholm has a permanent viking exhibit that is well worth visiting. You can also see some runestones from the viking era if you want to. There is one in the Old Town, embedded in a wall by Kåkbrinken-Prästgatan. Or you can make it a day trip: walk 15 minutes from Nordita to Östra Station, take the blue Roslagsbanan commuter train to Täby Kyrkby (about 15 min ride) and follow signs to "Jarlabankes Bro". Have coffee at Kardemumma and beer at Skänkstugan.