Until recently, only emerald, ruby, and sapphire were officially called precious gemstones. All other gems were referred to as semi-precious. The precious categorization was a reference to value: a really fine ruby, emerald, or sapphire can be priced higher per-carat than a diamond, which most people consider the ultimate gemstone. The reasons have to do with hardness, rarity, and beauty.
Sapphires are extremely hard and durable, with a very high color saturation and very few inclusions, so a fine, high color-saturated sapphire will look beautiful forever in a piece of fine jewelry.
Rubies, which are essentially red sapphires, since they are culled from the same mineral, corundum, are more rare, as are emeralds.
This makes rubies and emeralds extremely valuable, particularly if they are clean and highly saturated — but not too clean: rubies and emeralds without characteristic inclusions are suspect because they may be treated or synthetic. Inclusions in emerald in particular (called jardin) are perfectly acceptable.
According to the International Colored Gemstone Association, an emerald in a deep, lively green still has a much higher value than an almost flawless emerald whose color is paler. Emeralds are extremely rare. It is only seldom that a large specimen with good color and good transparency is found. That is why fine emeralds are so valuable. Note: both rubies and emeralds look their best in large sizes!
All gemstones are precious, and there are many, many exceptions to the big-three rule. Many rare finds among what we used to call semi-precious gems can be priced higher than the big three. But in general, a fine ruby, sapphire, or emerald will hold its value and command a higher price than other gemstones. That is the reason Art Deco jewelry — heavy on the big three — is usually prized over other period jewelry when sold at auction.