Please help me identify this plant

What plant is this?

I accidentally damaged their "mom" when replanting, some ranks came off, decided to put them in a shot glass, a week later, I was pretty surprised to see roots!

Also : how long should I let the roots grow before planting them in some soil? And what kind would be best?

(I hope you understand what I mean. I've never done anything like this with plants and English isn't my first language)