Why do some chemicals take electrons from other chemicals? How can some chemicals, if they have an equilimbrium of electrons, take away electrons from other chemicals?
One example I believe is placing a small amount of gallium on top some alluminum and watching the alluminum melt. 
Why does the gallium, if it is at equilibrium state, need more electrons? 
 A: 
One example I believe is placing a small amount of gallium on top some alluminum and watching the alluminum melt. Why does the gallium, if it is at equilibrium state, need more electrons?

No, gallium dissolving aluminium is not an example of electron exchange but of formation of an alloy.
Exchange of electrons between atoms in chemical reactions can roughly be explained by the Octet Rule. Atoms with less than 8 electrons in their outer electron shell (so called valence electrons) tend to shed or acquire electrons to make up a total of 8 (an octet), which happens to be a very stable configuration.
An example is the reaction involving aluminium foil and iodine crystals, which starts vigorously even at room temperature.
In it, $\mathrm{Al}$ loses three valence electrons, becoming an $\mathrm{Al^{3+}}$ cation, with the electron configuration of $\mathrm{Ne}$ which has a full octet.
The iodine atom $\mathrm{I}$ on the other hand acquires one electron, becoming an iodide $\mathrm{I^{-}}$ anion, with the electron configuration of $\mathrm{Xe}$ which also has a full octet.
The overall reaction is thus:
$$2\mathrm{Al}(s)+3 \mathrm{I_2}(s)\to 2\mathrm{AlI_3}(s)$$
Elements that tend to acquire electrons are called electronegative and are mostly found on the right hand side of the Periodic Table. Elements that tend to lose electrons are called electropostive and are mostly found on the left hand side of the Periodic Table. 
A: There is no electron exchange when you put gallium on top of aluminum. The known observed reaction is that aluminum diffuses into the gallium because it has a very high solubility there.
The tendency of an atom/molecule to take electrons away from others is related to the concept of electronegativity. See Electronegativity. In the end it is due to the fact that the total energy of the reactants is lower after the electron transfer. 
