When you're talking about briquettes, much of the "ash" you're talking about is actually clay and binders to hold the briquette together. That also tends to mean a cooler burning form of charcoal.
Overall, charcoal is made by taking wood (or a few other materials, but usually hardwood) and heating it to burning temperatures without enough oxygen for it to actually burn. The result is something that burns more cleanly and steadily than the original wood, which burns a fairly volatile way until it gets down to coal. That means that charcoal sort of "skips" the volatile burn and gets right to the steady cooking burn.
Briquettes are lots of little bits of charcoal, bound together with those binding materials into the little forms in the bag. Lump charcoal is just chunks of wood that go through the process and come out the other end still sort of resembling the piece of wood that went in.
Lump charcoal burns much hotter than briquettes and, since there are no binders or fillers, very little is left behind when you're done. However, it tends to burn out faster too, meaning for a long fire (like for slow smoking pork), you'll need to refill with lump charcoal more often than for briquettes.
Kingsford (which dominates at least the US market) is a company that comes directly from when Henry Ford really came up with the process for making charcoal. For most of that time, they've pretty much stuck to the same form factor with the binders, and focused on adding things like a few shavings of mesquite or pre-soaking them with petroleum-based lighter fluids.
However, recently, they've introduced a "competition" briquette that is actually much closer in how it burns to lump charcoal and has fewer of the binders in it. While I generally prefer to use lump charcoal, I wasn't nearly as disappointed when I tried these new briquettes.