In this tutorial, I will try to explain all options available when modeling in the polygon sub-object.
Under this you can see all the options you have in the PSO.
Shrink / Grow: Works exactly the same as in the vertex sub-object, only now it grows / shrinks by using polygons.
Insert Vertex: Use this if you really want a vertex at a certain position on a polygon. Just click 'Insert Vertex' and click where you want it on your mesh. Do notice the 3 or 4 edges it creates along with it! This is a good way to make your mesh ugly. Try not to use this. Better use 'Slice' and 'Cut'.
Extrude: Extrudes a polygon, either by it's own normal (perpendicular to the surface) or as a group. Click the do-it-yourself button after selecting the polygons you want to extrude and click-drag. Or do it 'by the numbers'. Then you have three extrusion types: Group (all in one averaged direction), Local Normal (as a group, but each polygon along it's own normal), By Polygon (each polygon along it's own normal and individually, so they won't form a whole). See the next image, where I first have selected all the polygons in the middle of a cilinder, and then applied one of the three extrusion types:
It's not like one type is the dominant one, they all have there purpose and I use them very often.
Outline: Scales the polygons you have selected (either by a certain amount, or manually). Could be useful, since it's not 100% a scale, since it scales each polygon, or group of polygons, so that they still have the same normals. Yet still, I haven't used it ever.
Bevel: Bevel is a combination of Extrude and Outline. So when you click drag you extrude, then let go and then when you move your cursor it outlines. This is actually a quite nice feature and I use it quite often (so that's why I never used Outline by itself! :).
Inset: Inset is about the same at Outline, with the difference that it creates new polygons, which is handy! So you'll be able to make polygons sit sort of inside others. This sort of operations can also be done with slicing, but for quick-and-dirty work, Insetting is the way to go.
Hinge From Edge: It does what it says: it hinges a polygon around an edge, so you'll create an attached prism... sort of. You can set the number of segments and the angle it needs to rotate when you do it 'by the numbers'. There you can also select the edge it needs to rotate over. Quite a nice feature which came in in Max 5 (people asked for it). I hardly even look at it. See the image below to get an impression of what it does:
Extrude Along Spline: This is a very, very fancy feature, which is handy when you try to model organic stuff, but also for technical modeling it could come in handy. Basically you draw a line, then click the button 'Extrude Along Spline', then click the line and it extrudes your initially selected polygons along that spline. Obviously drawing a spline with a certain shape is much easier then extruding all the time and trying to make it fit the line. Options such as Segments (the number of segments used to try and make it look like the line), Taper (to make it smaller near the end), Taper Curve (to make the smaller-going go linear or curvi-linear) and Twist (to give a sort of twirl / twisted look) all are very nice. It's actually more usefull than I thought.
Repeat Last: Don't understand why they bothered making a button for this, isn't there just a global undo button? :S
Create: With this you can create polygons, but not by a single click. You have to click a number (3 or 4) of vertices in the right order and then it will create a polygon for you. So for example, I clicked create and saw the first part of the image below (you will always see the vertices of your model when you click 'Create' in the polygon sub-object.
Click to enlarge
So here I first click the right-top one, then then the left-top one, then the left-bottom one, then the right-bottom one and then to complete the procedure, the right-top one again. Then it creates a polygon and selects it for you.
Collapse: Crashes your selected polygons into one point (vertex). Same as in the vertex sub-object.
Attach: Attaches other meshes to your current one. It's a bit of hardcore grouping.
Detach: The inverse of attach: make a selection of polygons, click detach and select the option you want. A complete Detach, so it will become a separate object, 'to Element', then you detach it, but it's still part of the same mesh, or 'As a Clone', then it copies the selection to a new mesh and keeps the old one.
Slice Plane: This is one of the most important functions in the Polygon Sub-Object. Select the polygons you want to slice (2), click 'Slice Plane', you'll see a plane gizmo appearing (3), rotate and move it until it slices your selection in the way you want it and click 'Slice', lo and behold for the result (4).
Click to enlarge
Slice: Performs the slicing as described above.
Reset Plane: Resets the Slice Plane gizmo to it's original position, extremely useful!
Quickslice: Quickslice let's you slice selected polygons, but dirty. It let's you 'draw' the plane that slices, by drawing only one line, so this is usefull if you would have wanted to slice while working from a orthogonal view. Do not use this for fine slicing, it's a very rough and very quick method. Works nice, but don't use it too quickly or actively.
Cut: With Cut you can draw edges onto your polygons (they don't have to be selected), you just click from point to point where you want an edge to be and it will create the necessary edges and polygons to do as you tell him. This is very handy for drawing shapes onto your model which you could extrude then for example. This tool is best used from already existing edges to other already existing edges, or you'll get the same problem as with insert vertex: it does what it's told to do, but along the way completely fucks up your mesh. Cut is a vital tool when modeling, try to understand how it works, and what it's strong and weak points are.
Tessellate: Tessellate is kind of weird at the beginning and I have only found one purpose for it so far. When you select a polygon and then 'Tessellate' it slices it two ways, exactly in half, so you'll get four pieces where you started with one. Would you press it again, then it would slice each and every one of the four newly created ones halfway, twice etc. So it's kind of a quickslice if you want to slice halfway twice.
Material ID: Used for texturing purposes (or selecting purposes) mainly. I am not going to cover texturing so I won't go into detail, but in short you can give polygons a Material ID which you can then match with a specfic material from a Multi-Sub Object material or use it when unwrapping.
Select by ID: Select the polygons that have a certain ID.
Smoothing Groups: Mainly used to create creases or pseudo-discontinuities on low-polygon models. By giving adjacent polygons a different smoothing group they get a hard angle between them. See for example here (left image with the center polygons with the same smoothing group, right one with different smoothing groups):
I never looked at this roll-out, until today and came to the conclusion that it's mighty handy. It basically is a built-in MeshSmooth in your editable poly, so you don't have to add the modifier. The advantage is that you don't have to add it (which makes your files bigger and more instable), the disadvantage is that you don't have a quick 'Show End Result' on/off toggle which you do have would you use the loose modifier. All the options are (almost) the same as the ones in the MeshSmooth modifier.
Use NURMS Subdivision: Click it to make things happen, basically an on/off switch.
Smooth Result: This acts as an additional non-mesh changing smoother. Basically a built-in additional smooth modifier.
Display: Iterations: Number of iterations (the higher, the denser and smoother your mesh becomes, be careful not to set it too high (>4) or else Max has a tendency to crash. This value can be different from the value used when rendering (see option below). Usually you set this lower, so as to make the viewport still workable but have good smoothing and a nice dense mesh when rendering.
Display: Smoothness: This is one of those weird options... Tweak it and you'll get a lower polycount, but you pay the price in the form of an ugly mesh, I don't really see why you would want use this option.
Render: Iterations: Same as the Display: Iterations, only at the moment you render.
Render: Smoothness: See Display: Smoothness, only at the moment you render.