They are Opuntias, but that is probably what you'll have to content yourself with, just the genus name. Opuntias are notoriously hard to ID, even if you have pics of fruit, bloom and pads. A few are distinctive enough to be able to be ID'd with just the pads, but most are not. Not being conversant in Opuntia-ese, I can't help you much.

Was this an outdoor plant hardy to your Z 8a? That might help you zero it down a bit. I can tell you what it isn't. Sometimes you need to wait quite a while for a good ID on them. The pads will change shape as it gets older too.

The pads broke off of a cactus at a local BBS. I tried looking online at the supplier's website, but they don't have pictures for every cactus and some of the photos were too "far away" to see the pads closely enough. >.Planto