Step 1: The cross
The first part of solving the cube is the cross. You solve 4 pieces on the bottom of the cube.
Again you can also watch the video from Arnaud on the right.
After the cross you have solved 4 pieces, to show which they are exactly there is the picture. 
Note that I only took out the rest of the pieces so you can see about which pieces we are talking now, it doesn't matter where ANY other piece is at this moment. Only focus on those 4 white edges.
Hold your cube with the white centre facing down. We start with the first step of the cross. To get all the white edges around the yellow centre first (or at least, the centre that is on the other side of white depending on the colour scheme of your cube). You can see that on the bottom picture. 
This is already a hard step, the white pieces can be everywhere still!
It takes a bit of practise but after a while I'm sure you can at least get 3 of the 4 pieces around the yellow centre
To get the last piece at the yellow centre I wanted to make some moving pictures here. Unfortunatly this is not possible for making the cross somehow. I will try to contact the maker of this so I can put the examples on later.
You will either have to puzzle a bit or watch the video one more time, although Arnaud explains it a bit different. My appologies for this!
After the first step
After you got all white edges around the yellow centre it's time to really solve them. You do this with 2 mini-steps. First you pick a random white edge from the top layer (so the layer with the yellow centre).
Move the top layer untill the side of the edge matches with a centre from the middle layer. 
You see that on the top picture, the orange edge is now lined up with the orange centre. This can be done in 1 move always.
You can now solve this edge by doing a half turn. It will end up at the correct place between the (in this case) orange and white centres.
Notation
To teach you already a bit of notation. On the picture you saw that the orange edge was on the right. I solved it with a half turn on the right side. In the notation this would be an R2 move. R for the right layer and 2 for the move.
Repeat this for all of the 4 edges and your cross will be finished!