When I think of an alive game, I usually think of Majora's Mask for the reasons people already mentioned, or Yoshi's Island.
In that game, it feels like you're really travelling through an island rather than progressing through sets of obstacles set up to thwart you. Bats in caves get bothered by you, you get caught off guard by birds dashing around, there's cacti playing catch with themselves, monkeys eating watermelons, monkeys getting to watermelons before you, and there's butterflies and flowers and dragonflies all over. It feels like you're in a part of that world, instead of just a stage in a game.
Even though there's clear goal posts, the stuff in between feels so organic and "already there" instead of there solely for the player character to triumph over.
That's not to say an empty sort of world can't feel alive. Mario 3 felt alive if for no other reason than there were multiple kingdoms. Rather than saying something like there's one way to make a world feel alive, the important thing is to remember what little tricks work and then making them work in whatever context the game has.
I'm kind of all over the place with this. My bad.