Is the enthalpy of a molecule determined by the energy of the electrons?
I find that enthalpy is one of the concepts that students struggle with the most and I feel your pain in trying to understand it!


  I understand that the strength of the bond of a molecule is determined by the potential and kinetic energy of the electrons


The strength of a bond is best understood in terms of potential energy. A strong bond involves a deeper potential energy "well" that electrons and bonded atoms can exist in. The atoms involved in the bond (and the electrons too) will have kinetic energy and that can explain how the bond behaves over time, but the overall intrinsic strength of the bond is a reflection of how low the potential energy of the system is at the equilibrium bond length. We call this the bond dissociation energy and it directly tells us how strong the bond is.

https://en.wikipedia.org/wiki/Morse_potential


  My textbook states that in a reaction where NaOH dissolves in water, the energy of the solid NaOH is greater than the energy of the dissolved ions in water. Is the "energy" of the reactants and products determined by the bond strength?


Yes, this refers to the potential energy of the system that is decreasing when the stronger attractions are formed in the solution vs the weaker ones broken in the undissolved ionic lattice and bulk water. Essentially, you have to break weaker bonds/IMF's in the reactants then you form in the products. Please note that this is a general statement and includes all "bonds" (covalent, ionic or metallic) plus other types of attraction (IMF's, ion-dipole forces etc.) that exist before or after a process or reaction occurs. 


  Also, my textbook states that enthalpy is "the amount of energy, in the form of either kinetic or potential energy, a substance has." I'm a bit confused about enthalpy. My textbook enthalpy changes:


The key part here is the potential energy. When this decreases (as in your example of NaOH dissolving), the potential energy turns into thermal energy. By itself, this doesn't change the enthalpy, as that is the sum of all internal energy at constant P. However, it will raise the temperature of the system and this makes it out of thermal equilibrium with the surroundings. Heat now flows from the system and this is when the enthalpy decreases as the system is losing energy. It is best to think of the enthalpy decreasing when the potential energy of the system decreases as the heat flow is what then tends to result. The reason this happens, as you are suggesting, is because stronger bonds are forming than existed before.


  Are these enthalpy changes determined by the bond strength also? And is the enthalpy of a certain molecule also from the bond strength?


Hopefully this last bit makes sense now, and the relationship between bond strength and the enthalpy of the system is a but less mysterious!!