Thanks all for your help! It is definitely an Echinocactus grusonii - the name on its tag just says barrel cactus, but it has those golden spines and yellow fuzzy top. I must also express that these can get 3 feet tall and wide I know :-). There is one that size in the C&S greenhouses at the Montreal Botanical Gardens. It's unlikely this one will get that big though - partly because I don't have a greenhouse for it, and partly because I am such a cactus newbie that I am unlikely to be that successful! So, I will plan on repotting it in the spring. Since it's in a mostly peat based soil right now I'll bare root it for a week as suggested. For now I will just place the plactic pot inside a terracotta pot to give it a bit more stability (it is quite tippy). It doesn't look like it is thirsty right now - it it very firm to the touch in the couple of areas I can touch it. One more thing: when you place the plant outside next spring, somehow mark the plant or container showing which side is facing south. Keep that side of the plant facing south all through the growing season, and the following year when you bring it back outside, make sure to orient the south side of the plant south again. I will do that, but I am curious as to why it isn't a good idea to turn the plant. Won't this encourage uneven growth? BP

Ok here's an additional photo of the barrel cactus in the process of repotting. It isn't the best quality photo but gives an accurate picture of what color each part of the cactus is. The brown color is not dirt; it is the actual color of the cactus. Please let me know if you think this is normal or if it is rotted. And if it is rotted, if you think it can be saved, and how. Thanks!!

Good you came back. All that mixture whatever it is needs to be got rid of, From the base of it and from the roots. The pot does look too large just as I suspected, cacti generally have small root systems. Barrel cactus is just a generic name, it gives no clue as to what it is. Seems like the gifter probably did know how to grow trees, but has a few things to learn when it comes to cacti :-) If there's no rot when you unpot it and clean off all that rubbishy mixture, it should be fine in the mix that socks has recommended. If there is, take pics and show us for further advise before repotting. It should be potted so that the bottom of its green part is level with the soil or ever so slightly higher, then you can top dress the soil mix with small grade gravel if you wish. That would stabilise the plant, make it look good and stop basal rot if the mix does stay wet....which it shouldn't! You can also mix gravel in equal amounts with the c/s mix and the perlite for a gritty potting mix, aquarium gravel, chicken gravel or horticultural/alpine gravel can all be used....but never, ever, use sand it's way too fine. If in doubt about anything come back. Gill

Those are called tubercles instead of arms. Unfortunately I see some dead tubercles on the side and the tubercles on the top have yellow bases which may be caused by rot or just not enough sunlight. I think the center top of your cactus may have rotted because the impact from the fall squished it. If the center top is rotten then I guess you could try cutting off the top and dusting the remaining base with cinnamon powder to discourage further rot but I think your cactus might be done.