The 4x4x4 octahedron is to the Rubik's Revenge what a Trajber's octahedron is to a 3x3x3 cube. It's about the same size as a Revenge and works just like its 3x3x3 cousins but with the extra layers.
The change of shape makes for a fascinating puzzle. The vertex tips are the equivalent of the 4 centres although, unlike the revenge, the position of the centres matters. These pieces are shaped like a rhombus. The tiny triangles are the equivalent of the corners on a revenge and the trapezium pieces are edges. There is complete choice of where you place each of the triangular 'faces' so that makes some parts easier to solve - you can often swap the triangular faces around like a 2x2x2 to set up a position you want.
The different shape makes it easy to get confused on this puzzle. You have to concentrate hard on what you are doing when you solve this one.
Although this puzzle can be solved with a Rubik's revenge method, it is tempting to try to exploit its shape to make it easier to solve. Before going for a straight revenge method, I messed around with pyraminx, 2x2x2 and 3x3x3 algorithms, including centre orientation algorithms. It's quite easy to build up at least half of the puzzle like this. It's also easy to end up with parity cases with so many equivalent pieces - including having 2 tiny corners that need swapping. It's easy to lose track with long parity correction algorithms. In the end, the way I solved it was with the cage method. The centre swap algorithm explained on the solution page for the revenge is actually a 3-cycle of centres and can solve the centres if you have all of the other pieces placed correctly.