What is a conjugate base and a conjugate acid?

It's just a relationship between a base and whatever happens to it if you add a proton, or an acid and whatever happens to it if you remove a proton.

Generically, it can written as:

#stackrel("acid")(overbrace("HA")) rightleftharpoons "H"^(+) + stackrel("conjugate base")(overbrace("A"^(-)))#

#stackrel("base")(overbrace("B")) + "H"^(+) rightleftharpoons stackrel("conjugate acid")(overbrace("BH"^(+)))#

So, what do you think? Is #"NH"_4^(+)# the conjugate acid or base of #"NH"_3#? What does that make #"NH"_3#?

Can water be both? i.e. Is water amphoteric? Why or why not?

The conjugate base of water, #H_2O#, is #OH^-#, of #HCl#, #Cl^-#, of #H_2SO_4#, #HSO_4^-#, of #HSO_4^-#, #SO_4^(2-)#. For each example I have simply removed #H^+# from the starting formula and conserved mass and charge in each case.

Likewise the conjugate acid of #H_2O#, is #H_3O^+#, of #HSO_4^-#, #H_2SO_4#, of #NO_3^-#, #HNO_3#.

In each case, I have simply removed OR added a proton (#H^+#) to give the conjugate base or conjugate acid.

What are the conjugate bases of ammonia, #NH_3#, and #HO^-#? These conjugate bases don't exist in water, but they do exist in liquid ammonia.