There's a thing called a "slug". " It is a mass that accelerates by 1 ft/s2 when a force of one pound-force (lbF) is exerted on it." (wikipedia).
Sometimes you'll see reference to a "pound-mass" to indicate a mass which weighs one pound at sea level (on Earth, thank you! :-) ).
The problem is that pounds and kilograms have been used colloquially since forever to describe the weight of objects. Scientific usage differs from informal usage such as "shipping weight".
EDIT: Though most of the comments have now been deleted, I state the following for completeness:
The kg is a unit of mass. Weight is a force, and like other forces it is measured (in the SI system) in Newtons. The weight of a body is given by the equation f=ma, where a is acceleration (in this case the local acceleration due to gravity.) Therefore we can rewrite this as f=mg
In the SI system, to say that something weighs 1kg is nonsense, because weight is a force, and should be quoted in units of force.
Nevertheless, there is a standard value for standard gravity g.
According to http://en.wikipedia.org/wiki/Standard_gravity:
9.80665 m/s², which is exactly 35.30394 (km/h)/s (about 32.174 ft/s², or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration.
According to the same reference, it is chosen as the gravitational acceleration at a latitude of 45 degrees at sea level.
This is of course absurdly precise given the global variation of g but there has to be an accepted standard value. It's also arguably not the most useful definition, since the majority of the earth's surface is below that latitude and of so it's likely to be slightly higher than what is needed in practice. If I had been given the opportunity to pick the value, I would have picked 9.8m/s² as a nice round number. http://en.wikipedia.org/wiki/Gravity_of_Earth shows that not just latitude but also continental landmasses influence the local gravitational acceleration.
So a 1kg mass weighs 9.80665N at standard conditions.
There is a unit called kgf. This unit is deprecated, because it complicates all kinds of calculations. Nevertheless it is an easy way for most ordinary people to visualise force, and like it or not, this unit is used in the real world. 1kgf = 9.80665N. If you buy a rope or fishing line from a local store, its breaking tension will likely be stated in kgf.
Let's look at some widely used units of pressure: Pa, Bar, atm, kgf/cm², mmHg, mmH₂O. The first two are based on the SI system. Numbers 2-4 (Bar, atm, kgf/cm²) are within a few percent of each other but all are widely used. The last three on the list all depend on the value of standard gravity in their definition. One reason for the perpetuation of units like mmH₂O is that they are easy to measure directly, for example in a water manometer. Another reason is that they simplify certain specific calculations, such as the design of water distribution systems.
When it comes to quoting quantities of gas by volume, the situation is even worse. People do measure gas in this way and it is important to know what temperature and pressure they are considering. If everyone could measure all quantities of substance in kg (mass!) and avoid talking about weight and volume, everything would be a lot less ambiguous.
Best Answer
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.