Extrude and Extract do almost exactly the same thing - so much that someday we are going to get rid of Extract and just make it a checkbox in Extrude. But in the meantime, you get two separate tools.
The Extrude tool takes your inputs selection, disconnects it along its boundary, moves it somewhere else, and then stitches the two boundaries back together with a "tube" of triangles. Extrude doesn't modify the connectivity of triangles inside or outside your selection - it just changes the boundary, and adds more in between.
The hotkey for the Extrude tool is d.
The property panel for Extrude is show on the right. The Offset parameter determines how far the extrusion is pushed away from the input surface.
The Harden parameter, when non-zero, allows you to insert a very small, narrow ring of triangles around the boundaries. This means they will appear as sharp edges even when in interpolated-vertex mode. This parameter dates back from before the addition of face groups, when hacks like skinny triangles were the only way to get the appearance of sharp edges. So, Harden may disappear in the future.
Density controls the number of subdivisions added along the extrusion path. This isn't a direct control, it is a relative slider, so 0 means no subdivisions and 100 means "lots".
The Direction drop-down controls how the extruded surface moves. The default is Constant, which means each vertex moves along its a constant direction, by the amount specified in Offset. The direction is defined by the average vertex normal over the selection. In the Normal mode, each vertex moves along its own normal. And in the X Axis/Y Axis/Z Axis modes, movement happens along constant directions in the world frame.
The EndType setting controls the shape of the extrusion cap. The default mode, Offset, is just a copy of the input selection. In Flat mode, we first do the Offset extrusion, then we construct a plane perpendicular to the offset direction, and move each vertex to that plane. An example is shown below. Note that this can easily introduce fold-overs and self-intersections in the mesh, so you may have to remesh the cap surface after the operation. Also, in Normal mode you cannot get a Flat endcap.
Finally, the Preserve Groups setting transfers the facegroups on the input selection to the output. If this is not enabled, then the facegroups are replaced with a new single facegroup. Note that the tube surface is always assigned a single facegroup.
The Extract tool is almost identical to the Extrude tool, except that instead of moving the selection and stitching with a tube, it just makes a copy of your input selection, and leaves it floating out there in space. This is useful for making layers. The hotkey for Extract is shift+d.