The 2x3x3, otherwise known as the Domino Cube, is a cuboid with the dimensions of 2 by 3 by 3. It turns a bit strangely, since you can only turn the R, F, L and B sides twice.
Example scramble (white top green front): L2 U' L2 U L2 F2 B2 U2 B2 U R2 F2
A fairly intuitive step. Find any white edge and simply put it next to its white center. Preferably, do this with white on the bottom. Where the edges are placed doesn't matter.
For example, in our scramble, we would do z2 L2 U' R2 to insert the two edges. If every white edge is in the white layer, all yellow edges will also be in the yellow layer.
In this step, we will use what I will call the right trigger. This is our first "algorithm" of sorts. It goes:
R2 U R2 U' R2
We will use this trigger to insert every corner. Find a corner with white facing up (making sure the white center is still facing down), and put it above any "empty slot", which is any corner with yellow facing down. Rotate the cube so this corner and its slot is in the front-right, and do the right trigger once. This should put it into that empty slot. Repeat this for all corners in the yellow face.
For example, in our scramble, we would do U y2 [right trigger] to insert one of the corners, then continue with U2 y' [right trigger] to complete the white (and yellow) sides.
Now, the two faces are complete, but the colours around the white and yellow pieces aren't. First, we will look for what we call "headlights" - where two adjacent corners have the same colour, similar to car headlights. If you find a pair of headlights, put them to the left and do this algorithm, which I will call the T algorithm:
[right trigger] y' R2 U' R2 U R2
This will solve the corners. If there are no headlights, do the algorithm anywhere and it should give you headlights. Repeat this step for the yellow face, and every corner should be solved!
For example, in our scramble, we would do [T algorithm], solving the yellow corners, then z2 U' [T algorithm] U2, solving the white corners.
Final step! All we have to solve now are the edges. What you can do is find two edges you want to swap, and use one of two swapping algorithms.
To swap two adjacent edges, use:
[right trigger] U2 R2 U2 R2 U' R2 U' R2
And to swap two opposite edges, use:
R2 U2 R2 U2 R2 U2
Use these two algorithms to swap edges until you solve the cube!
For example, in our scramble, we would do [Adjacent swap] y [Opposite swap] for the white layer, then z2 y' [Adjacent swap] y' [Opposite swap] to solve the cube.