It's a poor person's VR. Turns your phone into a VR headset. Ever sat on the couch and watched a bunch of YouTube videos? Or a TV show/movie? Put on the daydream, and you are sitting in a virtual theatre. It's good when you need the house to be quiet because everyone is in bed. I'd say 90% of my Daydream usage is watching stuff like Youtube/Netflix.
It's amazing when you are on a flight. It totally takes you away from the plane experience. Also really neat when travelling in general. Sitting in a hotel room watching content on a virtual big screen and not the pokey little hotel TV.
The cardboard is the super basic sibling to the Daydream. The Daydream has a hand remote controller that feels like a TV remote in your hand. It's how you control the phone while wearing the headset. The Cardboard is a bit limited because you can't really interact with it.