The swirling sun symbol can be seen when server is being contacted and things are loading. A few instances of when you see it is when you are saving, visiting neighbors' islands, on the neighbar or the game is just starting to load. The swirling sun may appear as a sign of Down for Maintenance, but not always.
Swirling Sun of Doom (SSoD)Edit
Players have taken to calling the Swirling Sun the Swirling Sun of Doom or SSoD because sometimes the game is stuck with the swirling symbol just swirling away. If this is happening to you, see the Improper Loading page for some suggestions for how to get in the game.