That is a cactus, looks like an epiphyte (grows on top of other plants). Make your cuts at the narrow junctions in between sections, for the least exposed surface area. Leave the cutting for a week in the shade, then pot it up and start watering after another week or so. You could probably lay a segment on top of the soil if you wanted to experiment but I usually go for the vertical orientation to start (plant obviously sprawls wherever it wants).
