I've seen some books use kg for kgf (kilogram-force), even though they shouldn't have conflated them.
But in this case it's not too harmful: if 25 kg means mass, as it should, then it the answer is direct. On the other hand, if 25 kg really means 25 kilogram-force, then the answer is the same under the assumption of standard gravity, because $1\,\text{kgf}$ is by definition $(1\,\text{kg})(9.80665\,\text{m/s}^2)$, the weight of 1 kg under 1 standard gravity.
The mass of the object always stays the same. The balance can only measure the downward force exerted on it by the bob. The force measured by the balance is simply the weight of the masses on one side needed to balance the downward force of the bob on the other side.
In air, the only appreciable force will be the downward force from gravity, aka the weight of the bob. In water, there is also a significant upward force due to the buoyant force exerted on the bob by the water. So in the water, the balance is measuring the difference between the weight of the bob and the buoyant force. The relevant physics and formulas can all be found on Wikipedia easily. If you define the (true) specific gravity $S$ as the ratio of the density of your bob $\rho_B$ to the density of water, i.e. $S = \frac{\rho_B}{\rho_{H_20}}$, you should be able to show that
$$ \frac{f_{water}}{f_{air}} = 1 - \frac{1}{S},$$
where $f_{air}$ and $f_{water}$ are the forces measured by your balance in air and in water respectively.
Regarding the use of grams or Newtons, they are often used interchangeably to talk about the weight of an object, although this is technically rather sloppy because they are not the same thing in general. The two units measure fundamentally different things, one is a mass and one is a force. However, since all objects on the Earth are subject to the same acceleration $g$ due to gravity, there is a natural way to change between one and the other, by the formula $f = mg$. Whenever people use grams to measure forces, or Newtons to measure mass, it is this correspondence that they are implicitly referring to.
Best Answer
The problem is that while mass is the same everywhere on earth, weight is not - it can vary as much as 0.7% from the North Pole (heavy) to the mountains of Peru (light). This is in part caused by the rotation of the earth, and in part by the fact that the earth's surface is not (quite) a sphere.
When you are interested in "how much" of something there is - say, a bag of sugar - you really don't care about the local force of gravity on the bag: you want to know how many cups of coffee you can sweeten with it. Enter the kilogram.
If I calibrate scales using a reference weight, they will indicate (at that location) the amount of mass present in a sample relative to the calibration (reference). So if I have a 1 kg calibration weight, it might read 9.81 N in one place, and 9.78 N in another place; but if I put the reference weight on the scales and then say "if you feel this force, call it 1 kg" - that is what I get. You can now express relative weights as a ratio to the reference.
All I need to do when I move to Jamaica (would that I could…) is recalibrate my scales - and my coffee will taste just as sweet as before. Well - with Blue Mountain I might not need sugar but that's another story.
So there it is. We use the kilogram because it is a more useful metric in "daily life". The only time we care about weight is when we're about to snap the cables in the elevator (too much sweetened coffee?) or have some other engineering task where we care about the actual force of gravity (as opposed to the quantity of material).
So why don't we call it "mass"? Well, according to http://www.etymonline.com/index.php?term=weigh, "weight" is a very old word,
Before Newton, the concept of inertia didn't exist; so the distinction between mass and weight made no sense when the word was first introduced. And we stuck with it...