From BlenderWiki

Subdivide

This is the simplest subdivision option, as it simply subdivides (“cuts in two” at the middle) all selected edges, adding necessary vertices, and subdividing accordingly the faces involved, following these rules:

• When only one edge of a face is selected, triangles are subdivided in to triangles, and quads, in three triangles.
• When two edges of a face are selected:
  • If the face is a triangle, a new edge is created between the two new vertices, subdividing the triangle in a triangle and a quad.
  • If the face is a quad, and the edges are neighbors, we have three possible behaviors, depending on the setting of Corner Cut Type (the drop-down menu next to the Subdivide button, in Mesh Tools panel):
    • With Fan, the quad is sub-divided in a fan of four triangles, the common vertex being the one opposite to the selected edges.
    • With Innervert (i.e. “inner vertex”), which is the default option, the selected edges are sub-divided, then an edge is created between the two new vertices, creating a small triangle. This edge is also sub-divided, and the “inner vertex” thus created is linked by another edge to the one opposite to the original selected edges. All this result in a quad sub-divided in a triangle and two quads…
    • With Path, first an edge is created between the two opposite ends of the selected edges, dividing the quad in two triangles. Then, the same goes for the involved triangle as described above.
  • If the face is a quad, and the edges are opposite, the quad is just subdivided in two quads by the edge linking the two new vertices.
• When three edges of a face are selected:
  • If the face is a triangle, this mean the whole face is selected – it is then sub-divided in four smaller triangles.
  • If the face is a quad, first the two opposite edges are subdivided as described above. Then, the “middle” edge is subdivided, affecting its new “sub-quad” as described above for only one edge.
• When four edges of a face (well, a quad!) are selected, it’s a bit special, as an additional vertex is created at the cross point of the two new edges linking two by two the new opposite vertices created on surrounding edges. This allows to subdivide the original quad in four smaller quads, rather than having e.g. four triangles and a rotated quad… much cleaner and simpler!

The example pictures should help you to understand all this better…

About FGons: when sub-dividing, FGons seem to be more or less converted back in their constituting quads and triangles, before the subdivision process itself…

The Subdivide Multi option found in the Specials menu (W) allows you to subdivide selected edges in more than two parts (the pop-up numeric field it displays allows you to specifies the number of cuts you want). This does not modify the rules given above, but may sometimes have strange side effects, with complex selections… Note that is you set the number of cuts to 1, this tool does exactly the same thing as Subdivide.

Options

The Subdivide tool (and its variants described below) has a few options, all only regarding quad faces, and located in the Mesh Tools panel (Editing context):

Subdivide

The “action” button, doing exactly the same thing as the menu entries…

Beauty

When enabled, the selected quads (i.e. those of which the four edges are selected) get a “special treatment”, as only two of the four edges will be subdivided (the two longest if Short is disabled, the two shortest otherwise). The same rules described above apply here as well, which means that for non-regular quads, the Corner Cut Type setting will also affect the result – see the example pictures below…

Short

Affects the Beauty option behavior, see above. It has no effect if Beauty is disabled.

Corner Cut Type

This drop-down menu controls the way quads with only two adjacent selected edges are subdivided, see above.

Examples

Here are a bunch of examples illustrating the various possibilities of the Subdivide and Subdivide Multi tools. Note the strange content of selection after subdivision… To enlarge a picture, click on it!

Basics
One edge, triangle and quad.	Two edges, triangle.	Whole triangle.	Whole quad.
Two edges in regular quads
Two opposite edges.	Two adjacent edges, Fan.	Two adjacent edges, Innervert.	Two adjacent edges, Path.
Misc for quads
Three edges, quad.	Three edges, non-regular quad.	Whole non-regular quad.
Beauty
Beauty, regular quad.	Beauty and Short, regular quad.	Beauty, non-regular quad, Fan.	Beauty, non-regular quad, Innervert.
Beauty, non-regular quad, Path.	Beauty and Short, non-regular quad, Fan.	Beauty and Short, non-regular quad, Innervert.	Beauty and Short, non-regular quad, Path.
Subdivide Multi (two cuts)
Triangle.	Quad.
Subdivide Multi (two cuts, with Beauty, non-regular quad)
Fan.	Innervert.	Path.
Subdivide Multi (two cuts, whole mesh selected)
Without Beauty.	Beauty, Fan.	Beauty, Innervert.	Beauty, Path.