$\ce{NH4-}$ is conjugate acid and $\ce{NH4}$ is conjugate base. $\ce{NH4-}$ is conjugate acid because it has one more Hydrogen.

The thing that is difficult for me to understand is why $\ce{NH4-}$ is anion? Is negative sign refer to whole molecule or just to Hydrogen? How a molecule can be anions?

Another example of the same situation is $\ce{HCO3-}$ (conjugate base) and $\ce{H2CO3}$ (Conjugate acid). By adding one Hydrogen to the base the negative sign goes away. Why?

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    $\begingroup$ You seem to have quite some things mixed up. I think you mean $\ce{NH_4^+}$ instead of $\ce{NH_4^-}$ and also $\ce{NH_3}$ instead of $\ce{NH_4}$. Maybe you should read up on acids and bases first before worrying about the cation/anion question. $\endgroup$ – Michiel Dec 9 '13 at 18:34

I'm quite confused about your post and even its title. So let's start wit title: ammonia $\ce{NH3}$ is neutral because of electron count. Now the $\ce{NH4-}$ is tricky. You would have to add hydride $\ce{H-}$ to ammonia. As far as I'm aware it's utopia. Hydorgen atoms in ammonia are protic so they would immediately react with $\ce{H-}$ giving $\ce{H2}$. I never heard that ammonia can react with metal hydrides as $\ce{KH}$ or lithium triethylborohydride $\ce{Li[HBEt3]}$.

Now to the text. Ammonia is amphoteric (i.e. it can get proton or loose it; or it can act as a base or acid).

Conjugate acid of ammonia ($\ce{NH3}$) is the ammonium cation $\ce{NH4+}$. Acid, in Brønsted theory, is a proton donor ($\ce{H+}$) and on loosing proton turns into base. In this case our acid is positively charged, so if it looses proton it turns into neutral molecule.

Conjugate base of ammonia is the amide anion $\ce{NH2-}$. If ammonia is an acid then to become base it must loose $\ce{H+}$. The final molecule is negatively charged. If you want to do it experimentally, you need to dissolve alkali metal ($\ce{Na}$) in liquid ammonia. H-radical is replaced by metal and dimerise to hydrogen $\ce{H2}$ (and salt - sodium amide is created $\ce{NaNH2}$).

And now your last question: "By adding one Hydrogen to the base the negative sign goes away". You are not adding Hydrogen (atom) but proton ($\ce{H+}$).

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