The cube has six faces. They'll be denoted:
We have 6 centers, 12 edges, 8 corners. Self-explanatory.
In a motion, if the letter is written as is, it means a clockwise turn. Otherwise, it's counter-clockwise. Also, usually I'll try to keep white face on top.
Just to get something straight - notice the centers of each side. Notice that these centers cannot move relative to each other - i.e. the white center will forever be accross the yellow center, blue accross green, and so forth. Therefore, if you place the red-white edge accross the blue-white edge, it'll never be correct.
That is why, we don't solve faces, but we solve layers. First we solve the top layer - white face while getting edges and corners correct (i.e. red-white edge faces the red center and so forth), then we solve the middle layer, and then the bottom layer.
First goal is to get the top cross. Forming the cross should be trivial, but you can run into two problems. First one is that you're not managing to get edge colors correctly. Solving this is quite easy:
Second problem is getting an edge flip. This too is easy to solve.
If you did everything correctly you should get what you see at the end of this picture.
Getting the corners is so easy, it makes me want to cry. Just locate a corner, see where it needs to go, prepare that place for it, and pull it in. This shows location of the orange-blue-white corner.
Notice that this assumed that the orange-blue-white corner is in the bottom layer. If it isn't, then it must be on top. So, what you need to do, is bring it down, by placing instead of wherever it is, an arbitrary corner piece. Also, it assumed that the white piece does not face down. If it does, then bring it in an out-of-order way to the top layer, then repeat what said above. Go on, and you should reach this:
Let's start with an example:
So, to get a piece, you first make a "T" with it (notice the blue "T"). Then you move it away from the side it needs to go. Then you bring down the upper corner, bring the bottom layer back and take it back up. Now you ruined a corner. Fixing it (with the trick we've seen before) fixes the whole thing.
Doing this repeatedly is enough by itself to get you through the middle layer. However, there is a shortcut! If you omit one corner in step 2, and you don't mind ruining it, you can do the following:
So, you can leave one top-corner unattended, do three middle-edges, then get the last top-corner, and get the last middle edge "classic way". Either way, you should end up with:
Notice: up to this step, all manouvers were understandable. From this step, it might be a bit more difficult to understand the reasons behind the steps themselves, and mostly you'll need to trust the manouvers and practice them.
Turn around your cube, now yellow will be up. Our first goal is to get a yellow cross, not necessarily matching to centers, but at least a cross. We're looking for the following:
We'll say that an edge piece is oriented if the yellow sticker is facing upwarsd. Looking at your cube, you'll be in one of the following cases:
If it's the first, then nothing is to be done - skip to the next step. If it's the second, here's what you do:
If it's the third, by doing the same manouver (B U L U' L' B'), you'll reach the second case (but make sure that the oriented edges are in F and B sides). If it's the fourth, the same manouver will get you to the third case. Nice, isn't it?
Now, you should have a cross at the top, but not necessarily all edge pieces are matching to their respective centers. Try to turn the top layer (yellow), and look to match as many edges as possible. There are three possibilities:
As before, if it's the first, then nothing is to be done - skip to the next step. If it's the second, here's what you do:
It might be useful to look at the blue-orange-white corner to see what's going on there exactly. If you look carefully, you'll see that what it ultimately does, is swap the U-F edge with U-L edge. Therefore, if you're in the third case, by doing the same manouver, you'll reach the second case. Thereon, make sure that you orient your cube right (matching edge pieces should be at B and R side), and do this step again. You should reach what the ultimate step of the previous thing looks like.
Our goal in this step is to get all of the corners in their correct places, i.e. the orange-blue-yellow corner willl be between the orange, blue and yellow faces, however we will not care about the orientation i.e. whether the yellow sticker faces up or not.
The following manouver, switches the edges: FRU stays still, and FLU -> BRU -> BLU. That is, the front-right edge stays, and the other three, forming a "triangle", rotate coutner-clockwise.
For me, the easiest way to remember is this "right, forward, left, forward, right, backward, left, backward".
Now, this showed a case where the trio-rotate was exactly what was needed. You might need to apply it twice. In any case, make sure to keep the good corner in the still place - FRU.
Despite being an easy one, this is by far the most confusing step. You have all the corners in place, but you need to orient them. What you need to do is really quite simple. You need to repeat the moves R' D' R D repeatedly, until they're oriented right. However, this'll screw some things up.
However, despite this, you will keep repeating this move, and moving the upper face in order to locate the "corner to be reoriented" always in the same corner (FRU in demonstration). Observe:
Et voila! You have your cube solved!