Looks like Tephrocactus articulates v, papyracanthus. Hopefully someone will confirm or correct soon!

Please note that they can rot easily if too much water. They need a very fast draining soil and sun. Difficult to say in the picture but your plant may need to be repotted with new soil and higher in the pot

Alain is right on with id and gave you very good advice. The mix needs to be faster draining. Just some plain ol perlite will do the trick. Mix 60% with some C&S soil. Your Opuntia needs way more sun. See how the new growth isn't as fat and round more long and narrow. This is a sign of not enough light. Most cactus do not make the best indoor plants. I'm not saying it can't be done, but most cactus are sun worshippers:).

I would start out by giving it full morning sun and then afternoon shade for a bit. Then see how the plant responds and up the sun exposure from there. Will not be happy if you give it all day sun all of a sudden.

Sorry I'm not trying to come off pushy. I just really like cactus and want them to all be happy:). Goodluck:).