Different people think different things about Minecraft, but at it's core, it feeds a need for some people to either (A) Survive, (B) Explore, (C) Be Creative.
That's it in a nutshell basically. It's not like other games with a directional arrow saying, go here, do this task, now do this... etc. It's different because it is open ended and there are no borders. You see a mountain off in the distance, walk to it and you're there. You see an island in the distance, swim to it and it's yours. You find a cave and want to explore it... go for it, it might take you 4 or 5 real world days to fully map it out, and that's not including any mining or resource gathering.
Also, the game keeps it simple. Find a basic resource like wood or stone, make tools... find better resources like Iron or Diamond, make better tools. Then once survival is no longer an issue and the food supply problem is fixed, you move on to improving your way of life. More storage, bigger houses and dwellings, some people farm so they build larger farms, some get creative with the game's physics and mechanics and build large machines or electronic devices, some get creative with the rules of animals and dangerous mobs that spawn, some stick to the purely aesthetic and search for rare materials to obtain rare dye colors to create their next piece of art... etc. It's all boiled back down to Survive... Explore... Be Creative.
For those who don't need a quest log telling them what to do next, it's the ultimate in sandbox / open ended gameplay experiences.
If you want to see more, check out this video that may explain it a bit better:
YouTube - Minecraft Fanmade Trailer
[video]http://www.youtube.com/watch?v=FaMTedT6P0I[/video]