Sterling silver refers to a type of silver. When you have a piece that is Sterling Sterling silver, or 925, you know that the purity, or more correctly the finess, 92.5% silver by mass. You can be pretty sure the remaining 7.5% is mostly copper. This “recipe” of silver has been around a long time and has been used in coins, jewelry, flatware, and so on. It is stronger then pure silver, or fine silver, which has which we say is 99.9% silver by mass, so we use it for jewelry or flatware and not fine silver.
Solid silver has to do with how the item is made. Sometimes an item will be made with one material then covered in a thin layer of an another layer. This is done for huge number of reason, however in the case of silver it is often, but not always, cost. One of the main methods of creating this “skin” of silver is a process called electroplating. When you only have this skin of silver, it can not be called solid silver. It can be called solid silver if it is solid sterling silver, fine silver, Britaninia silver, and so on. Solid silver does not mean always mean fine silver, although it is often used when talking about ingots and coins that are fine silver. Solid silver does not tell you about the qualities of the silver being used just that the item is made up of a solid piece.
When it comes right down to it solid silver means not silverplated! Since that plating is measured microns, fractions of the thickness of a sheet of paper, the amount of silver is meaningless in terms of value. So even a “solid silver” item that is a crude 50% silver by mass is going to have more silver then the silverplated item. When hunting down silver items finding the solid silver items is the goal. Identifing that something is Sterling silver, or another silver standard, will also tell you it's solid silver.