Your plant is an Aeonium hawarthii and is a member of the Crassulaceae family, a huge family of succulents that include many other popular and commonly grown succulents, including some that look a lot like aeoniums. Echeverias in particular are often confused with aeoniums and there are several other rosette-like succulents (eg. dudleyas, graptopetalums, pachyverias and graptoverias).
“One thing that sets these plants apart is the way their leaves attach to the stem- they are wrapped around the stem with a fibrous attachment so that when a leaf is pulled away, the stem is intact with only a transverse line showing where the leaf was attached. The other rosette Crassulaceas have succulent attachments and their being pulled off the stem leaves a divot in the stem.” https://davesgarden.com/guides/articles/view/1058/#ixzz2v0Z9rZjL
The existing plant looks to be quite leggy. As they are shallow rooted, these plants can get top heavy and may need to be rejuvenated. Aeoniums go dormant in the summer and will not root then so the best time to take cuttings is now or early spring.
Where rosettes exist, they can be cut with a couple of cms of intact stem. Leave the rosettes for 1 or 2 days to allow the end to callous before potting. You can also cut the stem into 5 – 8 cm lengths, long enough to insert 1 cm into the soil. Like the rosette,allow the stem to callus for a day or two before potting. A number of cuttings can be rooted in a shallow tray 3/4 filled with a well drained, gritty, potting soil topped with a layer of fine grit. Roots should form within 3 – 6 weeks. Note: Make sure that the stem pieces are planted in the same upright direction they were growing in. The plant prefers a Mediterranean-type environment neither too hot nor too cold so propagate in a place with plenty of light but not where the plants will be burned by the sun. Water sparingly so the plants do not rot.
For some further reading:
Succulents, The New Plant Library by Terry Hewitt (2002)
Succulents The Illustrated Dictionary by Maurizio Sajeva (1994).