Why should the redox reaction happen between the copper ions and iron atoms? I know that when $\ce{Cu^{2+}}$ ions react to $\ce{Fe}$ atoms, it becomes $\ce{Cu}$ + $\ce{Fe^{2+}}$. My question is, why?
Copper atoms normally have 29 electrons. This means that in the K-shell there are 2 electrons, in $L = 8$ electrons, in $M = 18$ electrons and in $N = 1$ electron. (Yes I know that there is a system with s and other letters, but I haven't been taught in that way at my school). Therefore $\ce{Cu^{2+}}$ should have 27 electrons, which means that the shells are assigned like this: $K = 2, L = 8, M = 17$.
So when the $\ce{Cu^{2+}}$ ions react with $\ce{Fe}$, why does it want to become $\ce{Cu}$? Because $\ce{Cu}$ doesn't have a complete electron shell at the end. Why doesn't it want to become $\ce{Cu+}$, because then it would have a complete shell at the end ($K=2, L=8, M=18$).
I know that when you take away electrons from copper, it does it in a special way. I'm 15 years old and not english so I'm really not an expert or anything, I'm just wondering why $\ce{Cu^{2+}}$ would want to become $\ce{Cu}$ when it meets iron ($\ce{Fe}$).
This is a good question, but you may have gotten off on the wrong track by thinking about the electronic shell structure (although what you have written seems correct, if a bit old-fashioned).  The relevant phenomenon here is the relative abilities of the metal ions to oxidize (or remove electrons from) each other.  For your specific example, the copper ion is a better oxidizing agent than the iron atom.  Therefore it is able to take two electrons away from the iron atom, oxidizing it and leaving the iron 2+ ion behind.  

As for why this happens, at your current stage of education the best answer is probably "because iron is easier to oxidize than copper".  You can look up the electrochemical activity series to find out how various metals relate to each other in terms of how easy they are to oxidize.  Copper is one of the harder ones to oxidize, but if you were to mix copper metal with silver ions, you would end up with silver metal and copper ions, because silver is even harder to oxidize than copper.

Does that help?