3D Modeling is the process of creating a model inside a 3D modeling program (such as Softimage XSI or Maya), for use in the same or other 3D modeling program. The goal of this process is very much what it sounds like - to make a model, or a representation of something, in 3D.
Modeling is the precursor to most other 3D work, such as rendering, animation, rigging, and so on. Without a model to work with, you can't do much.
There are a number of different technical methods, and dozens of techniques. The two main technical methods include NURBS and Polygons. Polygons are, today, the most popular.
How Polygon Meshes work:
One creates a mesh of polygons, which consists of three-sided (triangles), four-sided (quads), and sometimes more sided polygons which connect at the edges. Where the edges meet, there are vertices, and a polygon is defined by these. Vertices are (arbitrary) points in 3D space, a set of three coordinates for X, Y, and Z. Polygons connect to each other, and share edges and vertices.
Connected polygons form a mesh. This mesh can take any imaginable shape, and you can move vertices to modify it. You can also move polygons and edges themselves, but all this does (in most programs) is move all the vertices connected to that polygon or edge (which has the same effect).
The mesh can wrap around and attach to itself (like in a cylinder), but it doesn't have to, it can be left open. In most cases the mesh attaches to itself, because open polygon edges show that the model is only a paper thin skin and doesn't actually have any substance. It is, however, perfectly ok for a mesh to be open in a place you can't see it.
Usually a mesh is made up of almost exclusively four-sided polygons, with triangles to put it together where it won't fit.
Polygon modeling has the problem of making it difficult to make a smooth surface. In order to achieve a smooth looking surface with polygons, one must have thousands of them, much like pixels in an image. The problem with making this many polygons it that it becomes impossible to edit, and very unwieldy to work with. Instead of moving a few vertices, you have to move dozens of them to change the shape. Thus, subdivision.
Subdivision turns the actual polygon mesh into an invisible "control layer", and creates another finer "subdivided" mesh based on the original polygons. It takes every four sided polygon, and creates four four-sided polygons from it. Every triangle becomes three four-sided polygons. In short, it makes a mesh with more polygons.
The new, finer mesh doesn't fit directly on the original mesh. Instead, the program creates the subdivided mesh to be a smoothed out version of the original. It can be subdivided further, which again multiplies the number of polygons by four, and makes it that much smoother. With lots of small polygons in place of the original mesh, the model can be smooth.
In order to edit the mesh, one edits the original, unsubdivided mesh. The finer subdivision surface you can't edit directly. If you want to add detail into the model, you have to do it on the control mesh level. This goes for rigging, animating, texturing, and all of the other steps in modeling. You do everything to the original mesh, and subdivide to get a smooth model before rendering. This solves all sorts of problems, such as having a mesh too heavy to work with, and you are guaranteed a smooth surface.
Terminology is different in different programs. In Softimage XSI, any polygon surface can be a subdivision surface simply by pressing "+", which subdivides it. In Maya, a Subdivision surface is something between polygons and NURBS, but normal polygon surfaces can be 'smoothed', which does exactly the same thing as pressing "+" in XSI. When I talk about subdividing a polygon mesh, I'm talking about the "smooth" function in Maya.
Good and Bad Meshes:
This section will be a bit hard to understand if you haven't done any 3D work before.
There are in principle, two kinds of models: Hard body and, um, not hard body:
Hard body models are models that don't bend. On these models, you can do whatever you want, as long as it looks good. Functionality isn't a big deal, since as long as it looks good it's functional. Five sided polygons are fine, for instance.
Models that aren't hard body models, things like characters, cloth, are expected to bend. In order to bend properly, there are lots of rules you need to follow.
First off, five sided polygons don't bend well. The only kind of polygon a computer can actually draw is a triangle, (so a quad is really two triangles). It will let you work with quads, but everything that renders is, in the end, a triangle. This is true for NURBS meshes as well. If all goes well you never see the triangles. Turning a quad into triangles is relatively straightforward, the computer just has to draw a diagonal and it's done. A five sided polygon is a bit more complicated in this way. If you start bending the five sided polygon, it starts looking strange.
Second, is line flow. A "Clean" mesh is made of almost all quads. This makes the mesh much easier to work with (a mesh of triangles is confusing). Now a mesh of quads is (mostly) a grid, and the lines of that grid flow over the model. It gets more complicated though:
It's bad form to bend a quad diagonally, because of the triangles. What if the triangle edge drawn diagonally is going the wrong way? Most programs try to avoid this, but it happens anyway. Thus, in places where a model bends it should bend on the edges of the polygons, and not bend the polygons themselves.
On a characters face, line flow is especially important because the whole thing needs to be able to stretch. Good line flow is circular flow around the eyes and mouth, concentric rings of polygons. There have been endless discussions on the best possible line flow for a face, but in the end every line flow method is a compromise somewhere. Finding one that works is more important than finding a "perfect" one.
There are many more rules, and much more to making a good, working, polygon model, even though polygon modeling is actually much easier to work with than NURBS. Many concepts carry over to NURBS, such as line flow. NURBS were created in order to deal with the problem of not being able to work with a dense polygon mesh, but are now regarded by many people as obsolete, as creating and working with smooth polygon surfaces is now possible. Polygons offer much more freedom than NURBS, and are easier to deal with. The vertices of a subdivision surface work almost identically to the CVs of a NURBS surface. The only advantage of NURBS is that computers can redraw the surface faster after it is modified than they can redraw a subdivision surface after it is modified, and that old-timers in the 3D industry are used to them and still use them.