Compaction of sediments forces the sedimentary clasts to interlock. This is much like tamping gravel or sand around a fence post to make it more stable. Cementation of grains locks clasts together by precipitation of minerals in the pore spaces. Think of concrete, the lime is the cement that locks the sand and gravel particles together to form a hard driveway, sidewalk or foundation. This is the point at which the liquid can no longer hold disolved solids in solution.
Chemical reactions can also create substances that are insoluable in a given liquid.
These insoluable materials are also referred to as precipitates. For example, if you boil salt water until there is no water left, you will notice a crust of minerals on the bottom of the pan. This crust is a precipitate. In sedimentary rock terms it is an evaporite because, water was evaporated to leave the salty crust. The words sediment, and clast were mentioned above.