How much precipitation occurs for a given ionic product? Let us say we have a salt AB in water. Suppose the ionic product exceeds the solubility product at a particular temperature. Now, I know that the ions will precipitate out of the solution. My doubt is, how much of the ions will precipitate? By saying "precipitation starts", do we mean that all the ions will precipitate, or will some ions remain in the solution?
Solubility product is the product of concentration of ions that can be soluble / in aqueous ionic state under given set of conditions. When ionic product is more means, the remaining is not soluble. Only that amount precipitates out. Still there are ions in water, product concentration of which is exactly equal to solubility product.

For basic understanding of equilibrium constant, you can refer: http://www.adichemistry.com/physical/equilibrium/introduction/chemical-equilibrium.html