Silver, or Ag (from the Latin argentum) is a relatively rare metal that's a lustrous white. Due to its rarity, inability to rust in water, and low meting point/malleability, it has long been used as money, hence why it’s called one of the “coinage metals” alongside gold and copper. In RPGs it’s traditionally less valuable than gold, but historically this has varied. It’s a very soft metal, which makes it not very useful for making weapons out of. Despite this, it has a number of practical uses. Among these are that it has anti-microbial properties which has long given it a perception of being a holy substance, making it the bane of many monsters, and is why silverware and silver lined pitchers were things. It is useful in early mirrors, was a critical component of film photography, is still hard to beat for its passive anti-microbial uses, and continues to be used in some electronic applications. All in all, it’s way more useful than gold, with practical applications at multiple tech levels beyond just making currency and pretty objects.
Dungeons & Dragons
Like many RPGs, D&D uses silver as a currency worth 1/10th gold. Since adventurers make so much money, you’ll rarely bother with it after character creation. More importantly, many monsters are resistant to attacks not made with silver weapons. In 3rd edition, a weapon can be made of a magical silver alloy for an extra cost of 20GP (for a light weapon) to 90GP (everything else metal) and approximates silver’s softness by reducing the damage such a weapon does by 1. In 5E this costs a massive 100GP, but gold in 5E is mostly useless past character creation anyways since there’s nothing to buy, and has no further penalty. Ignore such weapons at your own peril, as inability to bypass such damage reduction is a major killer of low level characters.
Pathfinder, being essentially a 3E D&D clone, is virtually the same as the above. One critical difference is that only slashing and piercing silver weapons take the damage penalty, rendering a silver light mace an essential piece of adventuring kit.
Monster Hunter International
Virtually all monsters in Monster Hunter International are weak to silver, to the point they can’t even touch it without problems, with the only known exceptions being humanoids (which are generally non-hostile and die easily enough anyways) and a single quasi-werewolf created by a magic artifact.
Being a /k/ oriented work, MHI actually notes the inherent problem of silver ammo being unable to engage rifling and such speciality ammo making less reliable firearms choke. MHI and private hunting groups use silver ballistic tips on otherwise normal ammo. Government agencies instead get frangible rounds made with silver instead of lead, which put far more silver on target than the method used by private hunters. The very specialized nature of this ammo means the calibers silver ammo can be obtained in are actually pretty limited. Since sometime before 1983 (when the prequel series Memoirs is set) till the events of Legion (set in some January between 2010 and 2017), the only calibers available to private hunters (at least in the US) were .45 ACP, .308 Winchester and 12 Gauge buckshot. After Legion, obtainable calibers were expanded to include 7.62x39 and .300 Blackout, giving hunters access to much needed intermediate rifle cartridges. The preference for the "heavy and slow" side of ballistics (when modern trends are to the “light and fast” side) is an intentional detail: The amount of silver matters more for killing monsters than hydrostatic shock.
World of Darkness
Silver makes werewolves take aggravated damage they can’t soak. Silver is so anathema to werewolves that they will incur penalties from just a pack member carrying it. Fluffwise this is the price for Luna giving werewolf powers.
Some vampires, especially those abandoned by their sires, or those with strong Christian beliefs think silver can hurt them. This actually makes it hurt them, dealing aggravated damage. It is firmly established by Tremere experiments, though, that it's the power of their own faith that hurts them, not the silver itself and this weakness could be turned off with enough brainwashing via Dominate (or inducted in someone who didn't suffer from it previously, which is something Tremere like to exploit if they can get away with it).