1) Create an ellipse with te Ellipse Tool;
2) Copy it to the front (Ctrl+C+F/Cmd+C+F);
3) Reduce the size of 2) slightly (throught the Transform palette or the Object>Transform>Scale) and move it slightly, both using appropriate amounts;
4) Select both and Pathfinder>Divide, Ungroup (Ctrl+Shift+G/Cmd+Shift+G), and delete the unwanted parts.
A shortcut like Ctrl+C+F is made by holding Ctrl and then clicking first C then F, and then letting the Ctrl go.
1) Create an ellipse with the Ellipse Tool in the Toolbox (flyout from the Rectangle Tool), just click the tool and drag an ellipse, change the tool to the normal Selection Tool;
2) Copy it to the front (Ctrl+C+F/Cmd+C+F): using the shortcut creates a copy on top of the original, the copy will be selected;
3) Reduce the size of 2) slightly (throught the Transform palette or the Object>Transform>Scale) and move it slightly, both using appropriate amounts: in the Transform palette/panel, you may reduce the W (for width) and H (for height) values (which will make the copy ellipse smaller) and when you ClickDrag the copy slightly, it will cross the original ellipse in two spots; these two spots will be the sharp ends of the final path;
4) Select both and Pathfinder>Divide (this will create a group of separate paths with coinciding segments), Ungroup (using the shortcut Ctrl+Shift+G/Cmd+Shift+G) so that you can work on the individual paths, and delete the unwanted ones which will leave you with the one you want.
Actually, it will be easier to use the Pathfinder>Minus Front (Subtract from Shape Area in older versions).
And to get the gaps there are added steps.
1) Create the first ellipse;
2) Copy 1) and reduce the copy size, using Transfor palette or Object>Transform>Scale, then move as appropriate;
3) Copy 2) and reduce the copy size, for a uniform gap reduce W and H by the same amounts;
4) Select 1) and 2) and Pathfinder>Minus Front (Subtract from Shape Area in older versions where you need to hold Alt/Option to expand);
Basically repeat 2) - 4), only starting with 3), and moving in different directions (as in 2).