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In the image below:
$\ce{NH3}$ has no charge as it has a full octet.
$\ce{CH3}$ needs 2 more electrons to form a full octet, so has a positive charge.
Why does $\ce{N}$ of the $\ce{NH3}$ want to bond with the $\ce{C}$, even though $\ce{N}$ already has a full octet and neutral charge, and bonding would make it have a negative charge?
Is it because $\ce{N}$ is more electronegative than $\ce{C}$ ?
When working with nucleophiles and electrophiles, it is easier to think of them as "positive-ophiles" and "negative-ophiles" respectively.
With that said, NH3 (ammonia) has a lone pair of electrons that is attracted to positive charges. The carbon atom with three bonds is electron deficient and positively charged (which your image does not show). Thus, as a "positive-ophile"/nucleophile, the ammonia through its lone pair is attracted to and forms a bond with the positively charged CH3+ ion.
Yes, nitrogen does have a full octet, but its lone pair of electrons is nonetheless still attracted to the electron deficient and positively charged CH3+ ion. This is similar to how water molecules (also fully satisfied octets) can bind with positively charged sodium ions in solvation process of salt.
It's all about thermodynamics. The decrease in energy brought about by filling the carbon octet is larger in magnitude than the increase in energy caused by having a positive charge on the nitrogen atom (it is positive since the total charge of the system must stay the same). This means the sum of the energies of the products is lower than the sum of the energies of reactants, so the reaction is spontaneous.
