What Minecraft system is this?

Undoubtedly PC version. I suspect PS4/XB1 could run versions similar to this. Although this could be from a mod on the PC version.

The box-like things are barrels. They are available in a variety of mods, Factorization is one example. The Barrels Mod is another, might be a slightly superior example.

Barrels store many stacks of a single kind of item. For instance you can store 64 stacks of cobblestone in a single barrel from Factorization, but as soon as a single piece of cobblestone goes into the barrel it will accept nothing but cobblestone until it is empty again. Some mods have superior versions that can store even more stacks, and that's what those particular barrels are.

Deep Storage Units, added by MineFactory Reloaded, offers similar functionality but can contain a whooping two *billion* copies of that item. They don't have the nice graphic telling you what they are storing, however, and are slightly harder to get items out of.

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The pipes look like pneumatic tubes from Red Power 2 (or more recently, Project: Red, which is a continuation of Red Power 2 by an new author). They are one of the *many* item transportation mods for Minecraft. They are superior to Buildcarft's basic pipes, but pale in comparison to the much more powerful Itemducts from Thermal Expansion or the Logistic Pipes from Krapht's Logistics Pipes.

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All of the above mods are part of the Feed the Beast mod pack, which is what this gamer is playing, but it is a fairly complete item storage network.

Items come from below (where they are likely beign gathered by a Quarry) and then travel into the Pnumatic tube network. Since a barrel will only accept one type of item the items will move around the network until they find an barrel that is storing the item, at which time it will be put into that barrel.

If the creator of the network was smart, there will be a default destination for new items and/or a destination for items whose barrel is full.