Also my mother gave me a cutting off this plant. Id?

Mother's cutting is one of those grafted cactus with the red caps, don't know the cactus they use. They also typically look like that years later, LOL! I would look at Optunia as the first cactus (might be something else but check out Optunia family group first). Let them callous before potting them up. Most people say not to water them for a week after potting. There should be volumes on cactus cuttings here on this site. The cuttings look to be in pretty rough shape, and coming off a plant with black main body........ Worth a try though! Enjoy!

Could the second one (gift from mother) Hylocereus undatus a.k.a. dragon fruit - Pitaya - Strawberry Pear? Photo is very dark, close up would help.
If that's what it is, they get pretty large.

Thanks so much. Yeah That Main Body Was in rough shaoe. These come from a round about the have in front of their house. They've been flourish there even though they are left out all New England winter long and that's where the plow shoves all the snow & salt. They grab a few from there and planted then in their yatd. The ones they did last year bounced back and flowered this year. I'm going to cut them today and let them dry then pot them. The ones with roots have been potted already so I'll see how they get along. Worse to worse I can always go by and get a cutting from the ones in the round about which though being abused seem to be doing well. Thanks for the help

Snow, New England, defiantly sounds like prickly pare! Rina, do nurserymen use Hylocereus undatus for grafting the red caps on top of those? I had a grafted cactus where the red cap died and the green stem sent out these long trailing ratty looking branches very similar to the picture above?
Fred

I decided to try my hand at opuntia. I've seen people plant the prickly pear with cut end down, but I just harvested some from a neighbor's yard to start at home, and what I brought was a pad that had been laying flat on the soil with roots on the bottom and two sprouts/pups on the top, so maybe how the pad is laid in the pot is not that big a concern?

ez

Fred
They use few different ones and Hylocerus is one of them. Grafted cacti sometimes die after a while, and the rootstock will start growing, sending out branches and doing it's thing...happy it doesn't have to feed the graft :)

Here are some more used as rootstock:

Acanthocereus sp.

Cereus jamacaru

Harrisia jusbertii

Myrtillocactus geometrizans

Pereskiopsis spathulata

Selenicereus grandiflorus

Stenocereus pruinosus

Trichocereus pachanoi

Trichocereus peruvianus

Rina, you are a treasure trove of knowledge! Thank you Oh, do the grafted cactus ever survive? I've never gotten the graft to last.
Fred