Congratulations! You have a keiki (baby orchid). This sometimes happens when the mother plant is dying, but yours looks fine so it's probably just the genetics in the background of this hybrid.

Standard recommendation is to wait until the baby has at least a couple of roots that are two inches or so long before separating it from the mother plant - this may take a year or two. If the spike suddenly turns brown you'll need to remove the baby plant as it won't be getting any support from the mother, otherwise let it be until it grows bigger and has its own roots. Search this forum or google "phal keiki" as this topic comes up regularly. Good luck!

When the keiki (proper pronunciation: KAY-kee) has developed roots as Ephermerati mentioned, you can use a sterile razor to slice between the keiki and the flower stem. (If the connection between the two is really tight, cut deeper out of the flower stem rather than risk damaging the keiki.) You can dust the cut with cinnamon and allow the wound to callous overnight before potting the baby up.