Density Model
The simplest salinometer is a densitometer. Take an object with a known, relatively neutral buoyancy and put it in a graduated cylinder of a known volume. A very simple method is to take a soda straw and cap one end with a large lump of clay. Insert the straw in the water, with the clay end down, and it will float. As you add salt to the solution, the salinometer/densitometer will raise its float point. This type of salinometer is sometimes used in small salt-water aquariums. Its drawback is that it's not terribly precise.
Resistance Model
The resistance model salinometer uses the fact that salt added to water makes the solution a much better conductor of electricity. By sending a small current through a container with a solution in it and measuring how much current goes from the anode, or negative terminal, to the cathode, or positive terminal, you can determine how much was lost to resistance, and from that, derive how much salt is in the solution. Most industrial and portable salinometers work by this method.
Silver Nitrate Titration Model
This is the oldest method of determining salinity in a saltwater solution, and dates back to the Royal Navy and Robert Boyle. Take a measured volume of seawater, put in a small drop of potassium chromate, and then carefully measure in 0.05 cc drops of silver nitrate. The silver nitrate precipitates out the chlorine atoms; when there are no more chlorine atoms to react to, the silver nitrate reacts with the potassium chromate, forming a pink solution in the water. When the sample turns pink, note how many drops of silver nitrate were used. This will give a ratio of chloride ions to the volume of water, which maps back to the salinity percentage of the sample. Because resistance checking is faster and requires less equipment, and because silver nitrate and silver chloride are mildly toxic, the titration model isn't used as much.
Evaporation Model
While technically not a salinometer, it is also possible to boil off a sample of salt water of a known mass, and weigh the mass of the salts left behind. By comparing the two weights, you can derive the salinity percentage as a ratio. Normal seawater has 34 to 36 g of salts per liter.