Hello!
You might not now this, but Sedum is now split into many many genus. There are a lot of the genus, so not all of them are listed here. Here are the biggest groups that have been split into (Or just common ones): Phedimus, Petrosedum, Hylotelephium. There is also a few oddballs that have been moved to different genus.
Sedum (The "new" version):
The "new" Sedum genus mainly has the previous sedum that has rounded, pan-shaped/paddle shaped leaves, or ones that are chubby and juicy... I have them in two categories (Not official, just my personal thing I do..): Super chubby, and paddle shaped. Most of the chubby ones are not hardy (Such as the jellybeans, cleavatum, pachyphyllum etc. with exceptions of the tiny ones such as sexangulare or dasyphyllum, and Sedum album.) Nearly all paddle shaped ones are hardy. Most of those can also be grown as houseplants. Most of the chubby sedums are grown for leaves, nearly all hardy sedums are grown for leaves.
Many species like shade... some include: Ternatum.... Sarmentosum (Be aware when planting, it grows like crazy)
The "new" sedum don't have much dormancy, some still do though. They can be grown as houseplants.
*Do note that some species such as sexangulare and album, without aquedate sun, will become string-y.
Phedimus:
Phedimus grows like the groundcover Sedum.... Well, it just crawls about. Nearly all of them have scalloped leaves of varying degree. Most go dormant during winter, some do it very dramatically, others just stop growing. They are mainly grown for foilage and sometimes for flowers, but very rarely.
Phedimus does not make a good houseplants, with exceptions including most Phedimus spurius cultivars (Especially the variegated ones such as Tricolor).
Hylotelephium:
Many hylotelephium such as spectabile and sieboldii may resemble phedimus from their scalloped leaves, but they grow like bushes!! And also, they die back fully during dormancy. Some, such as sieboldii, look and grow like phedimus... But, usually, the leaf shape is much larger or rounder, and also, look for big Bunches of flower buds, I think it's called an Umble (??? Might not be correct term... ???). ANyways, in gardens you will mainly find Hylos for their flowers and leaves combined.
Hylotelephium does NOT make a good houseplant. It will get tired VERY quickly without dormancy. Mmm. *Note: It makes very good flowers in flower arrangements, they last very long and sometimes even root in the water...
Special oddballs (not all of them, there are too many.. here are some common ones...):
Sedum pachyclados:
Now in rhodiola.
