Is #H_2O# a base or acid?

Well... either-or. It's amphoteric.

Water can act as either a base or acid in the Bronsted sense. That is, it can accept proton (base) or donate a proton (acid), depending on the other species in the solution.

For example, ammonia is a weak Bronsted base, so when water dissolves ammonia, ammonia acts as a base, and water acts as a Bronsted acid:

#overbrace("H"_2"O"(l))^("Bronsted "color(red)bb"acid") + overbrace("NH"_3(aq))^"Bronsted base" rightleftharpoons overbrace("NH"_4^(+)(aq))^"Conj. acid" + overbrace("OH"^(-)(aq))^("Conj. "color(red)bb"base")#

(Note, however, that this equilibrium lies heavily to the left, since #"NH"_4^(+)# is a stronger acid than #"H"_2"O"#, with a #K_a# about 4 orders of magnitude larger than #K_w#.

The equilibrium lies on the side with the weaker acid.)

But, if water dissolves, for example, acetic acid (which is obviously a Bronsted acid), acetic acid acts as, well, the acid, and water behaves as a Bronsted base...

#overbrace("H"_2"O"(l))^("Bronsted "color(red)bb"base") + overbrace("CH"_3"COOH"(aq))^"Bronsted acid" rightleftharpoons overbrace("CH"_3"COO"^(-)(aq))^"Conj. base" + overbrace("H"_3"O"^(+)(aq))^("Conj. "color(red)bb"acid")#