Extrude allows us to pull a 3D object from a 2D shape. In its simplest form, it allows us to give depth to a 2D shape by extending that shape upwards for downwards. A circle, for instance, can be EXTRUDED to become a cylinder. A square can be EXTRUDED to become a box.
EXTRUDE - TO POINT
Moment of Inspiration allows us to extrude a shape in ever smaller increments until the shape comes to a point. Instead of a circle being extruded to a cylinder it can be extruded to form a cone. By default, EXTRUDE TO POINT operates at 90 degrees from the plane of the object being extruded. But, by selecting UNLOCK, the point can be placed anywhere in space.
EXTRUDE - SET DIRection
Rather than being restricted to 90 degree extrusions, Moment of Inspiration allows us to select a straight direction in which the extrusion will travel. Combined with ARRAY, this can be used to create some very interesting and complex shapes.
EXTRUDE - SET PATH
While EXTRUDE - SET DIR is restricted to a straight line, EXTRUDE - SET PATH permits us to draw a path which will be used to direct the extrusion's travel. This allows us to extrude more organic kinds of objects.
EXTRUDE - TAPER
This is a new capability in Moment of Inspiration. Currently, it is only available to owners of Moment of Inspiration that download the MOI 3.0 Beta. It permits the user to select an extrusion angle and distance. It's great for creating bases and expanding shapes.
Here is a tutorial that covers each of these EXTRUDE capabilities.