The previous posters seem to have provided pretty good explanations, but I wanted to add a few belated points on the purpose of lamination and Damascus steel.
Most traditional Japanese blade manufacturing involves welding a piece of high-carbon steel that forms the edge to a piece of low-carbon steel or wrought iron that provides support to the more brittle high-carbon steel. The hagane (which means "edge metal") of a cutting tool is usually a piece of steel that, when crystallized to the extent Japanese tools usually are, would be very rigid and brittle. Traditional Japanese blade technology addresses this problem by welding that hard steel to a body of the blade made of a less carburized steel or wrought iron (the "jigane") which is more ductile but won't hold a sharp edge. My understanding is that the softer metal absorbs the shock of sudden stress to the blade, so you don't shatter your chisel when you hit it with a hammer or break your knife when you drop it. The way in which the edge steel is welded to the body steel accounts for the terms mentioned: ni-mai and san-mai just mean "two-layer" and "three-layer", respectively, so obviously a billet of steel made for a double-bevel knife would have the edge steel in the middle of two layers of soft steel for the sake of symmetry and a single-bevel would have the edge steel on one side of a layer of soft steel. "Warikomi" means, as best I can tell, "wedged" or "inserted", which I believe refers to the method of splitting open or folding the soft steel to create a fold into which the edge steel is inserted. Interestingly, the techniques of forge welding hard steel to soft steel or iron were fairly common in western tool making before the advent of modern steel production methods that made steel more homogeneous and less brittle. Nowadays I suspect much of the purpose of having two different types of steel in stainless knives is largely cosmetic (especially in the "Damascus"-clad knives) but it does offer the possibility of having most of the knife stainless with a high-carbon edge.
"Damascus" steel is produced in a number of ways; I believe the beautiful nakiri pictured above, with its extremely regular, fairly thick layers, was made by welding several layers of two different types of steel together and drawing them into a bar--this technique is more accurately referred to as "pattern welding", and describes most "Damascus" steel produced with modern methods. The patterns of middle-eastern steel that were originally referred to as "Damascus" in the Middle Ages were made by drawing an ingot of crucible steel, which had pockets of different carburization levels because of the method of its production, into a bar. Medieval Japanese swords also have a similar, finer pattern in their steel (referred to as ji-hada). This grain is produced by reducing the irregularity and impurities in crude crucible steel by welding it to a piece of wrought iron and folding and welding the resulting bar several times. The refined steel would have been used as the steel for the edge of swords, rather than the softer core.
You can see the contrast between the hagane and the jigane on most of the knives pictured on CKtG. Apart from pattern-welded steels, most of the differences in the appearance of jigane, like "kuro-uchi" have to do with finishing rather than the metal itself. An exception would be the Itto-Ryu knives like this: http://www.chefknivestogo.com/rehafowh1gy2.html
, whose jigane is actually wrought iron (a form of iron that was ubiquitous a hundred years ago, produced by drawing pig iron out to create a bar with fewer inclusions and a linear grain structure, with almost no carburization). I'd spring for one of these in a heartbeat just because I love the way wrought iron looks, but they have an asymmetrical handle and I'm left-handed, so I'll have to settle for a moritaka aogami super (sad, I know).