Species name: Hylotelephium cauticola
Common name: showy stonecrop
Let me start off by saying that this was a pain in the tush to identify, having really nothing to go on. Sure, it looks a bit like a jade plant (in a related genus, Sedum), and sure, it looks a bit like hens and chickens (in related genera, Sempervium and Jovibarba) but it doesn't really fit any species in any genus that I looked at. An hour and a half later, I decided to look on the Dave's Garden website at all of the different cultivars in these genera and lo-and-behold! What amazing things human beings can do to plants once we start selectively breeding! This plant in its "natural" state is actually quite large, not the puny little plant that is dwarfed by the downspout in our garden. The flowers are also naturally a much lighter colour pink, almost white. This species of stonecrop (the common name "showy stonecrop" is actually quite misleading since it is applied routinely to about 5 different species in different genera) is native to Japan, with various cultivars being created and naturalized to North America. This plant does very well in dry, rocky conditions where most other plants would not grow, and is especially popular (with many of its relatives) as plants for rock gardens. It reproduces very easily from stem and leaf cuttings, including accidental breakage. There have been reports of pieces of this plant and related species taking root in a new spot in a garden after having being whipper-snippered off the original plant and having laid in the sun drying out for weeks.
Hylotelephium and the genus it used to be classified in, Sedum (DNA evidence suggests these are separate groups despite being incredibly similar visually), are both considered (along with many other plants) succulents. A succulent plant is a plant that has very fleshy leaves that are used to store water. If you have a jade plant at home, you'll probably have noticed that a day or two after watering the leaves look very plump and a bright green, and as time goes on the leaves get thinner and a much darker green (eventually becoming brown if you leave it long enough without water). This is the plant storing water in the leaf tissues, and gradually using that water for growth and basic baseline energy production as the earth dries. For this reason, it's actually detrimental to the plant if they're watered too often, and will very easily succumb to "drowning".
Other than the obvious ornamental value, this plant has no other medicinal or economic value.