what is branch water?

It's a southern expression. There are several terms for streams used in the USA and the names are mostly regional. In my area, the larger ones (that flow into a river) are creeks and the ones that flow into them are mostly called runs. To the south, they might be called branches. (We use "branch" for roughly equal-sized streams that form a creek.)

Anyhow, by extension, bourbon and branch is bourbon diluted by some water.

I am in the USA and I have never heard of it either.

http://www.answers.com/topic/branch-water

Addition of plain water rather than soda water to a mixed drink (for example, "Bourbon and branch" refers to Bourbon whiskey with plain water)

When a whisky is ‘cut’ (i.e. watered down) prior to bottling, the water that is used is very important to the final product. The preferred source of water is called ’branch water’. Branch water comes directly from the stream that the distillery is built on; some companies even bottle this water, so that bar customers can further dilute their bourbon with the original bourbon water. This branch water starts its life in the underground limestone shelf that exists under most of Kentucky and part of Tennessee. The limestone shelf acts as a natural filter for water that passes over it. Branch water is particular for its lack of character, with no traces of iron or other minerals that would be harmful to the whisky making process. Pedia for the wicked

http://en.wikipedia.org/wiki/Branch_water

you tell us

It is just another ploy for marketing

think of the story a bar keep might tell of the location of the stream or spring and how sweet the water is

Remember If you call a liquor or a mix you pay

Lots of my relatives would take day trips to the mountains to collect spring water and wait out the valley heat

I think it's just plain water. Although i'm from the uk too so i'm not 100%