Users become stronger, faster, more durable, etc. when they come in contact with crystals and gemstones, possibly unlocking abilities related to the affinity and enhancing the existing powers. Some users may be able draw sustenance from them or even slow or stop aging.
Chaos (Sonic the Hedgehog) becomes more powerful for every Chaos Emerald he possesses.
When armed with the infinity gems/stones, Infinity Gauntlet (Marvel Comics) gives the wearer cosmic control of the universe.
Powered by the infinity gem/stone of the mind, the Chitauri Scepter (Marvel Cinematic Universe) allows the wielder to control the minds of others, making them their obedient slaves.
Vision (Marvel Cinematic Universe) is powered by the infinity gem/stone of the mind.
Powered by the infinity gem/stone of time, the Eye of Agamotto (Marvel Cinematic Universe) is a powerful magical artifact created by the sorcerer supreme Agamotto with the power to manipulate time.
Saint-Germain, Cagliostro , Prelati (Senki Zesshou Symphogear AXZ) used Philosopher's Stones/Lapis Philosophorum in the form of gemstones to clad themselves in powerful alchemical armors called Faust Robe.
Chaos Emeralds (Sonic the Hedgehog)
Infinite (Sonic the Hedgehog) possesses great power due to the Phantom Ruby embedded in his chest.