Does anybody know what this plant is?
When we bought it, it was very small, brown and closed up. With decent watering and light it's grown into what you can see here: several low-lying rosettes of long thin leaves and a single straggling stem with the same leaves whorled around it.
Sempervivum tectorum, or Hens n chicks, grown under low light. The long stalk is the flower. Once the flower has finished the main part of the plant that the stalk came from will die and the "chicks" will take over.
The particular cultivar is almost impossible to verify as there are hundreds of types. The plant has been known or grown by people for thousands of years and is also known as houseleeks.
That looks like a Houseleek, Sempervivum tectorum, and it looks like it's going to flower. This is a succulent garden plant which grows with very low water and nutrition. It generally doesn't ever need to be watered in temperate climates when out of doors.
Your plant looks healthy, except for some signs of etiolation (growth deformed by insufficient light) and overwatering. Notice how the leaves become almost white at the bases, and on the flower stem, they are spaced widely. The plants will grow well in light shade, provided they stay very dry. When you keep them moist, and they aren't getting lots of sun, they will grow too fast, and become too soft, and pale. Your plant is mildly showing these signs.
Keep the plant in the brightest location you have available, and water sparingly. You may want to move it gradually, over the course of a week or two, to prevent sunburn.
Below are some pictures of what a healthy Sempervivum looks like.
