Inside the black box is a small, rose gold sphere with interlocking circles. It's called a Astronomical Sphere. Astronomical spheres, also known as Gemma's rings, are an early 16th - 17th century astronomical instrument. The instrument consists of three rings, each representing the celestial equator, declination, and the meridian.

What's cool is when it closes...

Kieran would have asked her, rather mysteriously, to "close it", so that once she did, she would see that the circles fold in and it closes into one solid ring band.

Basically, he is giving her his entire world. When the ring is open, it stands for the whole world, and when the ring is closed, it stands for everlasting love.