There are many different ways of going about the simple kitchen task of how to separate an egg, but I'm going to keep it simple show you just my two favorite methods.
Step 1: Choosing Your Eggs
I make an effort to buy free range, organic eggs when I can. Not only because I want what's best for my body, but because the yolks are stronger, which makes them easier to separate without breaking them. I'm not sure of the science behind why this is the case, but it's been my experience that it's pretty consistently true.
Also, I find that eggs fresh from the fridge have a stronger yolk than room temperature eggs. (Again, a bonus when trying to separate them.) So if you're following a recipe that calls for room temperature eggs, I would separate them fresh out of the fridge and then let them sit covered until they've reached room temp.