The two main things in minecraft are - you guessed it - Mining and crafting. It has a very staggered (and logical) method to keep you constantly upgrading your gear without feeling grindy. The deeper you dig, the stronger the materials you will find. Starting with wood on the surface, then stone, the iron and finally the elusive diamonds.
These aren't the only materials however. Virtually everything is accounted for, down to an electrical current that can be used to create elaborate traps and mechanisms (and even a working CPU inside the game itself).
Sure the combat leaves something to be desired, and even on hard difficulty you can build impenetrable fortresses, but minecraft is whatever you make it.
The creative will and have revelled in what minecraft offers, whereas those who prefer A to B linear games struggle to stay interested as minecraft forces you to make and set your own goals for the day.
The multi-player aspect of minecraft has its bugs and faults, but for the most part, it is just as much fun as the single player experience but with the added bonus of a friend (or as many as your server will hold) to help you along or hinder you.
Even if, by some miracle, minecraft doesn't have what you want in it, look to the modding community. Everything from new enemies, working villages (with NPCs and a trading system) airships, and an infinite number of things can be added (for free as well).
Final thoughts: Not for everyone, but if you like it, you will never forget it. Graphics: 9/10 (Blocks become astoundingly beautiful) Gameplay 10/10 Sound 9/10 Fun: 10/10 Replayability 10/10