The Trajber's octahedron was described as long ago as the 1980's. It is, in essence a 3x3x3 modification. If you look at some sites about modding, you will see that it is quite a common modification among the people who do this sort of thing. The puzzle in the picture is a factory-made one, QJ to be precise. It's quite small and a little fiddly to work with but less so than the shape-changing mods like the UFO.
The Trajber octahedron is a vertex-turning puzzle. It is a little like the corner-turning octahedron (way smaller) but doesn't have the trivial tips. Instead you have the little 'centre' triangles.
It's relatively easy to solve. I do 4 adjacent faces using the same approach as I would with a pyraminx. I make a slight adjustment to place the centres on these first 4 faces. I put in two long edges, then add the centre, join the centre to the final edge and insert the centre and edge together to finish the face. After this I remember that I am dealing with a cube. I turn the puzzle upside down and use sexy move to orient the edges of the bottom layer - I place the edges and rotate the centre if needed. Finally, I realise that the little triangles are not really centres but actually the equivalent to the corner cubies on a 3x3x3 cube. A couple of goes on a 3-cycle and the whole thing is solved.
I like the Trajber's. It's not as nice as the corner-turning octahedron in the way it's made or feels when you use it but it is an interesting puzzle with a decent history. The factory-made ones are quite cheap - about a fiver. Alternatively, if you are into building things, you could make one out of a couple of Rubik's cubes - there are guides on the WWW.