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Using your left-mouse-button, you can select components (vertices, edges, and/or polygons) to create an edge connection between them.
Middle-Click-Dragging will change the amount of connecting edge segments.
Holding Ctrl+Shift+RMB will display a marking menu that will allow for you to change the Middle-Click-Drag editing mode. There are three editing modes for Connect: Slide, Segments (default), and Pinch.

Note: Single edge selections will automatically connect the entire ring path. (less info)