I believe what you have there is a Tephrocactus articulatus var papyracanthus. Formerly classified as an Optunia.
I've never owned one but I would plant it in something that drains immediately and dries out in about 2-3 days which is what I do with all my succulents. In my area I use NAPA floor dry from the auto shop since it's made of sterile expanded diatomaceous earth, it simulates a mineral desert soil and only holds water for a few days. It's basically just like Shultz Aquatic Plant Soil (baked clay bits) but NAPA is way cheaper. I add a small amount of peat (10%) to the NAPA stuff to help it bind a bit like soil but not enough to hold excess water. Coir is good to use too in place of peat.
If you need to use peat or prebagged soil and don't have any NAPA, you can of course use perlite or small pumice but you have to use a lot of it (75%) to ensure the water drains fast and the pot dries out
entirely within a few days. If you stick your finger deep into the pot and your cactus/succulent soil is still moist after a week you need to come up with a soil mix that will dry out faster as they melt down in the dim cool days of winter. They literally turn into a gooey mushy mess like a over ripe piece of fruit when they stay too wet this time of year. I could never keep succulents for very long until I figured out my soil was the culprit. A soil that is always too dry is always better than too wet when growing these things. Don't waste time & money with those pre-bagged "Cactus Soils" they sell everywhere. It's amongst the worst thing you can pot a cactus in for the reason I just stated - stays far too wet. Unless you live in a hot, dry place, like a desert.
Here's the desert tropicals page for this Tephrocactus:
http://www.desert-tropicals.com/Plants/Cactaceae/Opuntia_papyracantha.html
Some google images:
http://images.google.com/images?hl=...articulatus+var+papyracanthus&spell=1&start=0
If your soil dries out in a the 2-3 day period I specified you can water perhaps once every 5-7 days in summer and maybe once every 3-4 weeks in winter but if your soil stays moist for a week or longer make sure the soil has been bone dry for a month or so before soaking it well again. Currently I'd say your plant looks fat and well watered. In winter I like to see them shrivel and maybe pull into the soil a bit before I water. Then I know they really need it.
In summer I fertilize with every other watering so every other week with bloom formula. You
do not want to use a balanced (10-10-10) or high nitrogen (20-6-6) fertilizer with succulents because it makes them grow soft and floppy. I use Dyna Grow bloom formula which is rated 3-12-6.