The beginners method for 2 by 2 has 3 steps. You first get the white side and the first layer, then you solve the second side, and finally you solve the hole cube. Shown below is everything you need to know on how to solve the cube. Everything is written down for you to memorize. Using this method and a good cube, you should be able to get under a minute when timing yourself.
The first layer is where you have 1 side solved and the colors around the side match up. Solving the first side and the first layer is usually the hardest step in the solve for beginners. When you start to solve the first side, you usually want to solve the first layer while you solve the first side. That way, you don't have to do a series of tricky moves to solve the first layer. If this does not make sense, press HERE or a video on how to solve the white side and the first layer.
There are 4 cases in this step except you only use 1 algorithm. This algorithm is surprisingly easy and goes like this: R, U, R', U, R, U2, R'. And the cases are shown below.
After you finish your first side and layer, you now hold the white side down. You look at the pieces on top and see which case you have. (cases are shown below) After you have identified your case (you may have to turn the top layer of the cube), you can do what that case says. After that, you will get another case and so on until your yellow side solves.
This is where if you are solving the yellow side and you have two yellow pieces on top of eachother. You are going to put them on the right side like shown. You do this by rotating the top layer, not by rotating the hold cube in your hands. Then you will do this algorithm and then you will get another case and you will keep going until you solve you second side. R, U, R', U, R, U2, R'.
In this case, you have 1 yellow sticker. You have to rotate the top layer, not the hold cube in your hands, until the 1 yellow is in the bottom left position. Then you do the algorithm R, U, R', U, R, U2, R'. You will get another case.
This case is when your 2 yellows are in a diagonal line. You can hold it anyway you want, as long as 1 of the yellows is in the bottom right position. To get them here, rotate the top face. Don't rotate the cube in your hand. Then, do this algorithm: R, U, R', U, R, U2, R'.
This case is when you have no yellow stickers facing up. You don't move from where you were holding it from. You don't have to do anything, except to just do this algorithm again: R, U, R', U, R', U2, R'.
In the last step there are 2 cases that you can get and 1 algorithm. If you get the unlucky case, then you will to the algorithm to get the better case. The 2 cases are shown below.
For this case, you hold your white and yellow side on the top and bottom, with the non white and yellow solved side in the back. Make sure that your layer is on the bottom when you do this. Then you do this algorithm: R', F, R', B2, R, F', R, B2, R2, B2.
For this case, you hold the white and yellow sides on the top and bottom. Then, make sure you hold the layer on the bottom while still holding the yellow and white sides on the top and bottom. Once your cube is in position, do this algorithm: R', F, R', B2, R, F', F, B2, R2, B2. You will get the first case. Then you can do what that says to solve your cube.