Silver Ore is a mineral obtained mainly underground, and more rarely above-ground. It is also commonly found on Floating Islands, much like Gold Ore. It is generally found in formations of 5-40 blocks.
Silver is one tier below Gold and one tier above Iron in strength and rarity.
In order to differentiate between Silver Ore and Stone, try to look for the separations that form between them, and note that the former sparkles while the latter does not.
To differentiate between Silver Ore and Iron Ore, note that the former is a lighter color.
Since v1.2, there is a chance that your world will have all of its Silver Ore replaced with Tungsten Ore during world generation, which will only allow you to craft Tungsten Bars instead of Silver Bars. Those two ore types may look different, but their functionality is the same.