The Corner-Turning Octahedron is built around a 3x3x3 mechanism. The vertices of the octahedron screw into the 3x3x3 core and the tips are fixed onto those. You turn the tips and the corner pieces giving you a puzzle that seems to be like a couple of pyraminxes stuck together. You can see how it turns in the image below.
You can solve half of the puzzle by twisting corners as you would on a pyraminx. If you solve a pyraminx easily, then you should be able to reach that stage with no difficulty. That leaves you with a maximum of 4 edges to position and untwist. This can be done with a couple of 3 and 4 cycle algorithms for positioning and a couple of algorithms to untwist the 2 or 4 edges that can be twisted at the end.
This is a lovely puzzle. It's very well made, looks nice and turns like a dream. It has a much more challenging relation in the Face-Turning Octahedron - make sure that you distinguish between the two puzzles when purchasing, they are are quite different in the way that they work. The puzzle in the picture was built from a DIY kit (for which no instructions were provided) and is manufactured by a company called Dayan.