Not a cactus but a Euphorbia, many of which look like cactus (but are in a different family) - convergent evolution in succulents is well demonstrated by them. That soil looks a little rich for it, but if it's growing well for you.....Anyway, Euphorbias are known to have a sap (milky) that can be a skin irritant or worse, depending on the person and the species of Euphorbia, so keep it away from your skin and all mucous membranes.

Ahh! Like I said, she had no idea what it was. It's in palm and cactus soil and it seems like it's happier than before we got it, but I guess that doesn't necessarily mean it's as happy as it could be. All of the leaves on the tips have grown since putting it in the soil, maybe two weeks ago, so it must not be too upset. What type of soils do these do best in? I never get succulents because I always kill them.