What type of reaction is this? How do you balance it? ##Hg(OH)_2 + H_3PO_4 -> Hg(PO_4)_2 + H_2O##

It's a reaction.

##3 Hg(OH)_"2" + 2 H_"3"PO_"4" -> Hg_"3"(PO_"4")_"2" + 6 H_"2"O##

This is a neutralization reaction between a strong acid and a strong base.

First off, you missed a subscript of 3, in mercury (II) phosphate; should be ##Hg_"3"(PO_"4")_"2"##

To balance it, you have to consider the subscripts and use coefficients effectively, so there is an equal amount of on either side of the equal sign.

##3 Hg(OH)_"2" + 2 H_"3"PO_"4" -> Hg_"3"(PO_"4")_"2" + 6 H_"2"O##

Hope this helps :)