$$\ce{NH_3 + HCl -> NH_4Cl}$$

The electronic structure of ammonia is such so that the nitrogen atom has an outer shell with a lone pair of electrons.

When ammonia reacts with hydrochloric acid (above), the hydrogen from the $\ce{HCl}$ moves to the ammonia. However, it leaves its electrons behind because there is already a lone pair of 2 electrons on the $\ce{N}$ atom, allowing it to retain a $\mathrm{1s^2}$ shell.

Why doesn't the hydrogen take its electron over and nitrogen doesn't then "donate" an electron to the chlorine?

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    $\begingroup$ This is an acid base reaction : strong HCl acid with weak base NH3 and it's a neutralization. Here the Cl- is behaving as a spectator ion. $\endgroup$ – bonCodigo Jun 2 '15 at 10:54

Chlorine is the more electronegative species. It would thus retain the bonding electrons when the hydrogen is evicted. This hydrogen then forms a coordinate bond with the lone pair on nitrogen.

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