What's the difference between binding energy and separation energy? My understanding of the two was as follows: the binding energy of a nucleus is, classically speaking, the energy needed to put together/take apart that nucleus completely (i.e. a measure of the strong force within that nucleus). However, in my mind separation energy is the energy necessary to take apart that nucleus into two (or more I suppose) specific constituent particles/nuclei.
I keep seeing these terms used, and I'm pretty sure my idea of these words is completely skewed. Could you tell me what these terms actually mean?
 A: I believe you have the basic ideas correct.
The binding energy is the energy required to create $Z$ separate protons and $N=A-Z$ separate neutrons from a $(A,Z)$ nucleus in its ground state.  Another way to think about it is binding energy is the mass energy which is missing from a nucleus compared to the mass energy of the individual nucleons.
When talking about separation energy one should specify what is being separated from a nucleus. One can calculate 1-proton separation energy, 2-proton separation energy, 1-neutron separation energy, etc. For example (as you suggest correctly), the 1-proton separation energy would be 
$$ [m(A-1,Z-1) + m(p) - m(A,Z)]c^2$$
where $m$ is the nuclear mass.
A: Basically, the Binding Energy is the total energy required to hold the whole constituents of the nucleus together. Whereas the Separation Energy is the energy required to remove a particular number of constituents(i.e neutron, proton or any other specified constituent of the nucleus) from the Nucleus.
A: Binding energy is energy required to completely disintegrate a nucleus into its constituent nucleons such that the nucleons don't feel nuclear force anymore. If we divide that energy by total number of nucleons we get average energy required to remove a nucleon which is around 8 Mev and is called binding energy per nucleon. However separation energy is energy required to remove only one nucleon at a time from nucleus e.g for calcium (42) it is around 11.48 Mev (ist seperation energy in a way similar to ist ionization energy). If we keep on removing nucleons one by one, time will come when only two nucleons will be left in nucleus and energy required to remove the last one will be 2.224 Mev. If we add all these seperation energies and then divide by number of nucleons,  we will get back the binding energy per nucleon. The difference between the two ways of getting binding energy per nucleon is that one is time average and another is ensemble average.
