What exactly is binding energy? I've been reading on radioactivity but along the way I got confused, if binding energy is the amount of energy used in holding the nucleus together then why is binding energy also the amount of energy used in breaking the nucleus apart, shouldn't the energy required to break the nucleus apart be greater than the initial energy?
 A: This is very basic: we are looking for the minimum energy supply that we must apply.
This is very similar to that thing students use to ask: "Why do we need a force $mg$ to overcome weight? It should be greater". And, okay, rigurously speaking, if you apply a force $mg$ upwards, you won't rise the weight. But you will have cancelled the weight force, so the object will now behave like a free body.
So, if you apply the tiniest increment, say $10^{-9}$ more newtons, for example, that will be enough to rise it, because now the body is free. But I could have also chosen $10^{-12}N$, and the object would also go upwards. The acceleration would be really small, true, but that would be enough to rise it.
So, we obviously need to give a number (not any than I can think of, but an objective figure). We choose to say that you must apply $mg$ to cancel weight, and then, the more you apply, the more you'll get. It is understood that it will have to be greater, but how greater is up to you. We just give the lower limit: the one that just cancels it out.
So this is the same. If we choose a free body to have $0J$ energy, then any bound state has less than $0J$, (negative energy), because it hasn't got enough energy to reach freedom, so it must be less than $0J$.
Let's call that energy $E_b$, and $E_B<0$, so the body has energy $-|E_b|$. You must apply $+E_b$ to overcome that. Like that, the particle would be able to reach freedom (probably at infinity), without any KE. If you want some extra $KE$, that's the extra energy you need to supply.
A: You are on the right path, but it needs a little explanation. Let's go from the smallest known particles up. 
Let's say that the neutron is made of 3 quarks (it's actually made of a sea of quarks, antiquarks, and gluons, antigluons, the net is called valence quarks), those quarks do have rest mass, right? Sure. Add them up, and you get 1% of the rest mass of the neutron.
Where does the 99% of the rest mass of the neutron come from? It is the binding energy between the quarks, called color force.
Now let's go one size up. Lets look at the nucleus. It is made of neutrons and protons. The rest mass of the (separate) neutrons and protons will add up to 101% of the rest mass of the combined nucleus. 
Now how is that possible? How can the constituents rest mass be more then the bound system made of the same constituents?
The answer is binding energy between the neutrons and protons called residual strong force, or nuclear force. But why does that decrease the rest mass of the bound nucleus?
From wiki:

Nuclear binding energy is the minimum energy that would be required to disassemble the nucleus of an atom into its component parts.
  The mass of an atomic nucleus is less than the sum of the individual masses of the free constituent protons and neutrons

If you want to tear a nucleus apart, and that is nuclear fission, you need to add at least as much energy as the energy that binds the constituents together. When you reach that energy limit, you will get a peanut nucleus. What the energy limit is, depends on how much energy binds the nucleus's constituents (neutrons and protons) together. That energy is the residual strong force or nuclear force. When you add enough energy (meaning you add a high energy neutron), that is at least as much as the binding energy, the nucleus will fission into two separate daughter nuclei. First, it will be a peanut nucleus, and at that point, the EM repulsion between protons in the daughter nuclei will be enough (it will overcome the nuclear force) to shoot the two daughter nuclei apart at 3% the speed of light.
