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So what I know is, a salt is an ionic compound between a metal and a non-metal which exchange electrons. Like in $\ce{NaCl}$, $\ce{Na}$ is the metal and $\ce{Cl}$ is the non-metal, $\ce{Cl}$ hogs up the valence electron of $\ce{Na}$ and so the atom becomes charged.
But in the salt $\ce{NH4Cl}$, neither nitrogen, hydrogen nor chlorine are metals. All of them are non-metals. How can it be a salt?
This is not the correct definition of salt. The IUPAC definition of salt is "a chemical compound consisting of an assembly of cations and anions". In fact there are many salts which do not fit the definition you give.
If we take the Lewis view of acids and bases then it is clear that ammonia has a lone pair which can be donated to a Lewis acid. The proton is the Lewis acid in this case.
Thus ammonia and a proton are able to react to form an ammonium cation. If the proton was obtained by breaking the H-Cl bond in hydrogen chloride (this can either be done by dissolving hydrogen chloride in water to form a dilute solution of hydrochloric acid or by the gas phase reaction of an amine and hydrogen chloride) then the anion required for electroneutrality of the system will be the chloride anion.
If the water concentration is very low then solid ammonium chloride can form while if plenty of water is present then the ammonium and chloride ions are well solvated by the water and exist as a solution of ions in water.
It is interesting to point out that ammonium chloride can decompose on heating to form ammonia and hydrogen chloride. When the vapours cool down then the ammonium chloride will reform. This causes ammonium chloride to be able to sublime with ease when heated.
