The X-Cube is not a cube - it isn't even a regular polyhedron. You can see this "cube" here.
At first glance, the X-Cube appears to be a subset of a 5x5x5 cube, and one might assume that techniques for solving a 5x5x5 would apply equally well to the X-Cube. However, the 4 outer faces of the cube may only be rotated when the outer face is complete. Given the cube's asymmetry, many 5x5x5 manipulations will rapidly create a state from which outer moves are impossible (but would have been fine on a normal 5x5x5 cube). Clearly another approach is necessary.
One important observation is that the X-Cube can be treated as a 3x3x3 cube with extra outer faces attached to 4 of the 6 natural faces of a cube. There are 6 "inner" moves (Front, Back, Left, Right, Up, Down) which operate on this 3x3x3 core just like a normal cube. If you restrict yourself to these inner moves and mentally ignore all the extra stuff protruding from the core, then you can work the X-Cube just like a 3x3x3 cube. It may look a bit strange in the middle of a solution, but as long as no outer moves were performed you'll be able to restore the X-Cube with your favorite 3x3x3 algorithm.
Now let's talk about the "extra stuff". In a solved X-Cube, 2 of the faces are shaped like a cross (see the illustration above), while the other four faces are 3x5 rectangles. Note that since the centers (and their colors) are fixed, the orientation of the colors is also fixed and thus the same two colors will always be cross faces, while the other four are outer faces. On my cube, Yellow and White are cross faces, while Red, Blue, Green, and Orange are outer faces.
Each corner of the core has two outer corners attached to it, and this pair of outer corners shares the same three colors. On of these colors belongs to a cross face, the other two are outer face colors. Core edges are a little more complicated. 8 of these reside on a cross face and have a single outer edge attached. We'll call these "single edges". Four of the edges lie between outer faces and are attached to 2 identically colored outer edges. We'll call these "double edges".
Notation
Orient the cube with the cross faces to the front and back. Normal cube moves will refer to rotating the cross faces (F, B) or the inner moves of the other faces (U, D, L, R). Outer moves will use lower case letters (u, d, l, r). One center slice operations is also somewhat useful: H will stand for rotating the horizontal middle layer clockwise when viewed from above. This carries the right center to front, front to left, left to back, and back to right. Note that this operation is the only one that changes the orientation of the fixed centers.
It is often useful to refer to a core piece and its associated outer pieces as a single group. When the group is referred to as X, then the core piece itself is X0, and the outer piece(s) are X1 and X2.
The notation X/Y refers to swapping pieces X and Y. Similarly, X/Y/Z is a cycle that carries X to Y, Y to Z, and Z back to X.
Solution Overview