What is the weight of an object of mass 1kg on the earth?

The weight of an object is the gravitational force exerted on it. It is different on different planets, as the gravitational force on every planet vary.

It is given by:

$\text{w}=\text{mg}$ 

Where g is the acceleration due to gravity on the earth.

$\text{m}=1\text{kg, g}=9.8{\text{m/s}}^{\text{2}}$

$\text{w}=1\times 9.8{\text{kgm/s}}^{\text{2}}$

Since weight is a force, its SI unit must be Newton(N).

$\mathrm{w}=9.8\text{N}$