The weight of an object is defined as the force of gravity on the object and may be calculated as the times the acceleration of gravity, w = mg. Since the weight is a force, its SI unit is the newton.
You might well ask, as many do, "Why do you multiply the mass times the freefall acceleration of gravity when the mass is sitting at rest on the table?". The value of g allows you to determine the net gravity force if it were in freefall, and that net gravity force is the weight. Another approach is to consider "g" to be the measure of the intensity of the gravity field in Newtons/kg at your location. You can view the weight as a measure of the mass in kg times the intensity of the gravity field, 9.8 Newtons/kg under standard conditions.
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At the Earth's surface, where g=9.8 m/s2 :
The kilogram is the SI unit of mass and it the almost universally used standard mass unit. The associated SI unit of force and weight is the Newton, with 1 kilogram weighing 9.8 Newtons under standard conditions on the Earth's surface. However, in the US common units, the pound is the unit of force (and therefore weight).The pound is the widely used unit for commerce. The use of the pound force constrains the mass unit to an inconveniently large measuring unit called a "slug". The use of this unit is discouraged, and the use of exclusively SI units for all scientific work is strongly encouraged.
What unit is force of gravity measured in?
The force of gravity is measured in the same units as any other force. A few
popular ones include newtons, pounds, dynes, and stone. ('Kilogram' is not
a unit of force.)
It's not possible to describe the nature of gravity on one planet or another
in terms of force, because the force depends on the size of another mass
placed on that body.
The nature of the gravitational field in the vicinity of some large body is described
in terms of the acceleration of gravity there. In SI units, it's meters per second2.
In English units, feet per second2. The value is 9.8 (32.2) on Earth, and 1.62 (5.32)