Wood Framed Construction
Building with wood is probably the easiest method of construction - at least for me. Some new technology (not really new) that is gaining popularity due to decreased install time is stressed panel systems. These systems essentially sandwich 3.5" of foam between 2 pieces of plywood. It's rigid insulation therefore provides better insulating values compared to fiberglass batt insulation.
Pros: Renewable resource, easy to construct, forgiving material, well-known
Cons: Insulating value, air infiltration, termite and insect infestation
Another method of wood framed construction is Timber Frame. The timber frame home uses heavy timbers (6x6 and above) to frame bents that are typically 8' on center. Between the bents are infill materials of typical wood framed or squared logs. Probably the best quality of timber frame construction is the aesthetic look of the timbers and the joinery used.
Steel Stud Construction
Steel stud framing is probably the second most used method of framing for the residiential market and probably first is commercial projects. Much the same variables apply with steel as in wood framed structures. You still have the same materials applied to the exterior of the building. As far as the ease of construction - it's no different than wood framed. Light gauge steel construction has one different than wood framed - insects can't eat it. Other than that the only other major difference would be that screws are used to put together steel instead of nails.
Insulated Concrete Forms will probably begin to take over the housing industry before too long. Don't get me wrong - there will probably always be wood framed and light gauge buildings. The concepts behind ICF make it a far superior structure than any wood framed building. Structurally - ICF is stronger. Air Infiltration? - What air infilitration? One problem with ICF is that people (homeowners and contractors) do not understand how to design for that tight of a system. It'll take some time - but will get there. For more information you can ask questions on the Forum
Log and Timber Construction
Log and timberframe is probably the oldest method of construction. Some of the major concerns with log and timber frame is the air infiltration and maintenance of the structure itself. Most of these concerns can be dealt with new technologies such as SIP (structural insulated panels). These are most often used with timberframe construction for walls and roofs and for log roof structures.
There are many more types of construction but these are the major ones at this time. Research is happening all the time in the construction and related industries.