How do you separate ships

Click on a fleet, then select a ship. Choose the 'Eject ship' command from the tab.

When you click on a fleet as Director suggested above, note that you can also hold ctrl key while you select ships and then "eject ship" will cause all the selected ships to be ejected as a new fleet.

You can also select the Manage option which brings you to the fleet management screen. This is useful for regrouping ships into separate fleets when dealing with many ships in one hex.