First of all, weight and mass are not really the same thing to be stringent. When talking about weight of an object it is in terms of physics the gravitational force the Earth is applying on the object that we are measuring. However, a pair of scales at home will show you your weight in kilograms or some other unit of mass and not Newtons. Confusing? Therefore, this section will only deal with mass. For forces, the that section.
The kilogram is the SI unit of mass and is defined as the amount of mass of a particular piece of object stored in Sèvre, France acting as the International Prototype Kilogram. Most units are multiples of the kilogram with exception of the carat which is often used for measuring gemstones and pearls and the atomic mass unit, also known as dalton (Da), which is approximately the mass of an nucleon. The exakt definition is a 12th of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state.
The avoirdupois system is a measurement system based on a pound made up of 16 ounces. It is the common system used in the United States and is to some degree used in the UK and Commonwealth. To make things more confusing there are alternative systems with units with similar names but of different magnitudes. See this article for more on this.
The troy units are commonly used to precious metals and gemstones. Unlike avoirdupois, there are 12 troy ounces in one troy pound, with similar differences for the other units.
The apothecaries units is a historical system of mass units that primarily has been used by apothecaries and physicians. It is closely related to the troy system.
The metric system introduced in 1924 has replaced the traditional Japanese Shakkan-hō (尺貫法) system but for example the momme is a recognized unit in the international pearl industry.