There are several popular ways to create Polygon Models. No one way is the right way and most modelers become proficient in all of them. Very often it is about personal choice or even about how you were taught or which books you read when learning the craft. This section give you an overview of the main ways to create geometry and an insight into where to use the technique.
Where possible I have added link to places where you can see these styles in operation either in a book or training video.
3.01 Point by point
With this method of modeling you create geometry by adding points to create polygon faces then move the points to the correct place by tweaking them. In some programs like Lightwave you can copy and paste points which keep them roughly in the same plane.
Some programs have tools that allow you to interactively add points by selecting a point on an existing model then sequentially adding more points until a new polygon face is created.
With this method you can build up a model in patches or islands then stitch them together with a connect/merge/create polygon command. This type of modeling lends itself to using background images as reference in various viewports.
3.02 Edge extrude
Similar to point by point, this method involves selecting edges and using an extrude command to create new polygons. Starting from a single polygon you can very quickly build up a detailed models.
3.03 Box modeling
For this popular type of polygon modeling it is usual to start off with a basic primitive shape (often a box hence the name). Using a variety of tools (connect, split polygon, knife, split face loop, etc.) the modeler defines the shape of the model very quickly and with very few polygons. Once the overall proportions are defined it is time go do another pass splitting polygons, spinning edges, connecting/merging points (and a plethora of other tools) to get good edge loops and get nearer to the desired shape. It is not as easy to work with background reference images for this type of modeling
Sculpting with brushes is one of the newer ways of creating flowing organic polygon models.
Pixologics ZBrush and Mudbox are brush based freeform organic modeling packages. They have alpha brushes, 2D and 3D tools all designed to push, pull, indent or elevate geometry. ZBrush also has ZSpheres (and more recently Dynamesh) that gives you the ability to join spheres together with a central link (which is actually more spheres) and then covert it to a very regular geometric model when needed.
3DCoat uses an organic modeling system called voxels. Voxel sculpting allows you sculpt without any topological constraints and make intricate details without thinking about the geometry. You can also change topology as you wish. It is not based on surface deformation but on volume building and filling.
MetaBalls have been around for a long time (two geometric shape that organically blend into one another but they never really created good edge loops or regular geometry.
There are occasions when a model needs to be created with very accurate topology by using a high resolution mesh and re-creating the new topology over the top. This is often the case when a model has been created in a sculpting package like ZBrush or 3DCoat.
The high resolution mesh can can from scan data or from a program like ZBrush that can handle multi million polygon meshes with ease (RAM permitting of course). This method gives you great control over where you want your edge loops to lie and allows you to localize detail by simply drawing it onto a very disorganized mesh in a very organized way
Silo3d uses a Topology Brush. This allows the modeler to use a high resolution base model and draw new lines that will become new geometry when finished
Other software with this sort of topology technology includes ZBrush, 3DCoat, Topogun and XSI.