Insoluble salts are ionic compounds that are insoluble in water: the salt continues to exist as a solid rather than dissolving in the liquid. On the atomic scale, the ionic lattice of an insoluble salt remains intact; the ionic lattice does not break up to allow the salt ions to be surrounded by water molecules and so form a solution.
If an insoluble salt forms by the reaction of soluble substances in water and falls out of solution, we call it a precipitate (see image).
When a salt such as sodium chloride (table salt) dissolves in water, its ionic lattice is pulled apart so that the individual sodium and chloride ions go into solution.
NaCl (s) → Na+(aq) + Cl-(aq)
In practice, many salts that are described as insoluble do actually ionize slightly in water, releasing ions into solution.
The undissolved solid in contact with water will come into equilibrium with the ions it has released. At this point the solution is said to be saturated.
In simple cases, where there are no common ions or competing equilibria, the ion concentrations depend only on the equilibrium constant for the particular salt.
When we talk about solubility equilibria we always write the equilibrium with the solid on the left. For example:
Ba(IO3)2 (s) 
Ba2+(aq) + 2 IO3-(aq)
The equilibrium constant expression for an insoluble salt is written following the same rules as for any other equilibrium. The equilibrium constant is called the solubility product, Ksp. The Ksp expression for the above equilibrium is:
Ksp = [Ba2+ ] [IO3- ]2
You can learn more about solubility products here.