"Crafting" games are those in which the player combines multiple resources into higher-value items. The key challenges for the user are opportunity cost (a resource used to build X cannot be used to build Y.)
Goal: survival (build shelter and fend off baddies), efficiency (produce lots of X, or the best-cost solution for X), wealth/achievement (make lots of X)
Progress directed (fixed path or goals), sandbox (free placement and open decision-making)
Actors: first-person, character, directed agents, third-person omniscient ("god mode")
Environment: 2-D surface ("top down" or isometric view), 2-D plane ("side scroller" or "platformer"), 3-D
Example games I have played:
Don't Starve: Survival, Sandbox, Character, 2-D isometric
Played this a while back. Didn't get very far into it.
Infinifactory: Efficiency, Directed, First-person, 3-D
Very fun game about setting up production lines. Not a lot of choice in what you produce, but plenty of freedom on how to do it.
Triple Town: Wealth, Directed game with Sandbox metagame, Third-Person, 2-D surface, mashup with match-3
This one is a bit of a stretch. Unlike other crafting games, while you can only combine multiples of the same element, and combinations are formed via placing groups of three or more.
Factorio: Survival (though I play with enemies disabled), Achievement (build a spaceship), character but with heavy automation, 2-D surface
This is all about setting up large-scale industrial infrastructure. Achievement is not enough to make me actually finish the endgame. Sandbox is pretty fun, though, since the goal is to build machines to build things, not just craft everything yourself.
Craft the World: Survival, Achievement, Sandbox, Directed Agents, 2-D plane
Currently playing, sort of a 2-D Dwarf Fortress, of which there are many options. May be done soon, been through one endgame, and the different environments are not enough to keep me hooked.
Terraria: Survival, Sandbox, Character, 2-D plane
Dwarf Fortress: Survival (doomed!), Sandbox, Directed Agents, 3-D world (with 2-D views only)
Really, I frequently ask myself why I'm not playing this more. The NetHack of this genre with many object interactions.
Puzzle Craft: Wealth, Directed, Third-Person, 2-D plane, another match-3 mashup
Definitely a crafting game, the main game is used for collecting resources, while the metagame involves building the village, but in a very strictly controlled manner.
Swords and Potions: Wealth, Directed, Directed Agents + Third-Person, 2-D isometric
Create a shop, build up the town with items made there, to produce more raw materials. Sell to adventurers, or outfit them to send on quests. Kind of grind-y but I would like to find a game like this with more depth.
The problem is when I look at the new games coming out in this category, they all seem to be concentrating on some axis I don't really care about. Some are science fictional instead of fantasy. Many are trying for better 3-D graphics or incorporating some more role-playing mechanics. But very few seem to have a unique goal, or a unique mechanic, that makes them at all interesting to me.
About the only one I considered was Blueprint Tycoon which is a stripped-down efficiency game sort of like Factorio.