First let me say that I have over 20 full fly boxes divided between trout, steelhead, and saltwater, eleven or twelve for trout fishing alone, so "organizing" them practically requires a flow chart.
Steelhead is easiest, because I do the least of that, and fly selection is typically not very complicated. I have two of those typical SA-type plastic, foam-lined boxes, the long ones. One box holds smaller low- and medium-water flies, darks on one side, brights on the other. The other holds the big boys, marabous, speys, and rabbit strips. Pretty simple.
The saltwater stuff gets a little more complicated. I've got eight boxes, six what I call "boat boxes" and two "beach boxes." The boat boxes are the clearish plastic, shallow, tray-style tackle boxes with removable dividers. I usually let two to four flies rattle around in each individual bin. Three of the boxes are pretty big, about 11"x17", and the other three are about half that size. I carry them in a tackle bag designed to fit them stacked, with pockets for reels, tools, and sundries. One of the big boxes holds chinook, lingcod, and other "big" flies like articulated deceivers, giant clousers, and jumbo flying squids (my own pattern). The second holds bright clousers and other weighted attractors in sizes for cohos and pinks, and the third holds epoxied and other unweighted baitfish imitations, surf-candies, sea-habits, deceivers, etc. The three smaller boxes are divided between rockfish flies (mostly weighted half-a-rabbits), mini-baitfish patterns, and small attractor patterns for staging fish. The beach boxes (which usually go into the boat too) are the same type of boxes, but small enough to fit in my chest pack. One holds euphasids and other krill paterns, small attractors, and chum flies. The other usually carries a limited assortment (drawn from the boat boxes) of bright and imitative clousers, and weighted and unweighted baitfish flies, all in a couple sizes, depending on what I think I may encounter that day.
OK, trout; now it gets real complicated.
I try to divide them into lake boxes and stream boxes, with some overlap. I have a standard size plastic foam-lined boxes for mixed stream nymphs, and another for buggers, leaches, zonkers and other streamers (this box does lake and stream duty). Dry flies and emergers I carry in six small boxes, organized by bugs for lake or stream: caddis (2); pmd/baetis; March browns/drakes; callibaetis; midges. I use these really cool little boxes made of clearish white plastic and divided into six small compartments, each with its own hinged lid. They cost about $4; I think Kauffman's carries them. I also have a largish Perrine aluminum box with individual lidded compartments on one side and clips on the other, sort of a low-rent wheatley. It carries big dry attractors and bead-head soft hackles in the compartments, and all my chironomids in the clips (this breaks my lake/stream segregation, but chironomids are the only flies that the clips don't ruin). I have two of the small plastic boxes for lake flies, one for damsels, adults on one side, nymphs on the other, and the other carries, waterboatmen, scuds, and other lake nymphs. Finally, I'll usually also have along a couple old prescription pill-bottles, film canisters, sucrets boxes, or some such full of whatever.
I know it all sounds pretty sprawling and not particularly efficient or "organized", but I almost always fish out of a boat, so I have the luxury to spread out a bit.