OK, a mat run is for ship materials and they come from hunters (regular mats) and warships (rare mats). You start getting mats after you fill the cargo hold and trigger the hunters who start chasing you and which you sink. Each hunter gives one regular/common mat chest and sometimes other loot, which may or may not stay in the cargo, depending on what type of loot it is. After a short period of time sinking hunters, warships come after you as well. At this point, you have both hunters and warships chasing you. For every warship you sink, you get a rare ship material chest, and sometimes other loot as well, just like the hunters drop. You can sink both hunters and warships until the cargo hold is filled with ship mats and loot skull chests. If you sail until full, you'll only get ship materials (rare and common) and any skulls you may have acquired. All other loot is forfeit, such as cargo crates, loot pouches, royal chests and loot chests (excepting of course loot skull chests, which stay on board). There will be some gold left for ya, but it aint much.
A loot run is where you don't sink any hunters or warships, but keep sailing until your cargo is full of the best chests, be they skulls, royals or regular loot. You can sail as long as you like, but eventually your cargo hold will fill up with loot skulls, which means you're full and cannot acquire any more anything.
Typically, you'll spend well over an hour if you're trying to fill up on skulls only, and it's almost impossible to do because regular ships will randomly start dropping ship materials, which fill in one of the slots in the cargo hold of the ship.
The goal here is to get loot and gold and avoid ship mats.
There are tiers of loot, where cargo is the lowest and is dropped over board in favor of better loot. Treasure chests are next lowest and also get dumped. Next are loot pouches, which will also get dumped in favor of better loot. Next come loot chests, which will also eventually get dumped as better loot comes along. Then come royals, which carry gold, very little gold. Royals get dumped for either loot skulls or any type of ship mat.
Loot Skulls, regular mats and rare mats never get dumped, and when yer full of them, you can't get anything else.
I hope I confused you well enough...