The Skewb, invented by Tony Durham is a corner-turning twisty puzzle, consisting of 8 corner and 6 square center pieces. This puzzle is a Pyraminx shape-mod, working on the same 4 axes mechanism and this is why it was first called the Pyraminx Cube. If you take it apart you realize that it has the same inner cores and the four corners are actually centers and the other four pieces are edges.
The Skewb and its bigger brother, the Master Skewb
At a first glance the corner-turning Skewb might seem to be unusual for those who are only familiar only with the Rubik's Cube, but it's easy to get used to it and it's actually easier to solve than a 2x2x2 Pocket Cube. After the first face is done it can be solved in maximum 24 steps. It is very fun to play with, no wonder why it became an official WCA event. The current World Record for the fastest solve is held by Jonatan Klosko with an amazing 1.10 seconds.
There are many Skewb mods: Skewb Diamond (photo) where the corners become centers, the Pillowed mod, the Ultimate Skewb where you have to orient every piece, but there are dodecahedron shaped mods, and Siamese Skewbs, and the list goes on.
It's highly recommended to buy a Skewb because it's an easier WCA cube, they usually have a great quality with clicking mechanism and it makes a great collection piece.
In this solving method we are using the classic FRU Rubik's Cube notation but in this case the letters mark the corners instead of the faces. We assign the letter F, R, L, B, U to the corners of the cube as seen on the image. A letter by itself means a clockwise ↻ rotation and an apostrophe marks a anticlockwise ↺ turn. The Skewb being a Pyraminx mod a turn of a corner means a 1/3 (120o) rotation.
Hold the puzzle in our hands with the corners facing you!
1. Solving the first face
Try to bring the matching corners aroun a center piece. Use this stage to familiarize with the puzzle and try to do this without reading any help and algorithms. This step is similar to solving the corners on the first layer of the Rubik's Cube so knowing that is a big advantage.
When three of the four corners are solved but the last one is oriented wrong use one of the algorithms below to rotate the highlighted piece. This algorithm won't harm the solved pieces on the top of the cube, marked with arrows.
Clockwise ↻: F R F' R'
Coutnterclockwise ↺ (direction on the image): F' L' F L
You should see a complete face at the end of this step. Note that the color of the top corners on the sides match, but it's not important to fit to the side center pieces yet. Unsolved pieces are marked with grey stickers.
2. Cycle the centers
Now that the first face is done, we have to put all the remaining 5 centers to the right place. To do that we can use an algorithm to cycle 3 centers around a bottom corner. Take the cube in your hand so the solved face is up and do the following as many times as necessary.
Rotate around F clockwise ↻: L B L' B' - R' B' R B
Around F counterclockwise ↺: R' B' R B - L B L' B'
3. Solve the remaining 4 corners
Here's a simple algorithm which rotates the bottom corners of the Skewb. It turns the two corners on the right counterclockwise ↺ and the ones in the left clockwise ↻.
Hold the puzzle with the solved side facing up. If there are two bottom corners that have both the color of the bottom center piece on the same lateral side of the cube (orange headlights on the image), then move that face to the front-right side and do the algorithm below. If you can't find two such pieces then do the same algorithm until you can see one.
L' F L F' - L' F L F'
When you master this solution you might find interesting the page about solving the Skewb blindfolded.
Solution of the Skewb Diamond
I mentioned before the Skewb Diamond mod where the corner pieces of the Skewb become centers and the centers become corners. This puzzle is an octahedron having 8 sides and the ones facing each other having the same color. You can buy a Mefferts or Mozhi version, they both move very well after lubrication.