A carat is a unit of weight used to measure the size of a gemstone such as a diamond. A karat is a measurement of purity, indicating the proportion of gold out of 24 parts (24k gold being pure gold). 18K gold is therefore 18/24 parts or 75% gold.
Size
Gemstones have different densities, so a one carat diamond may be a different size than a one carat stone in another material.
Alloys
Pure 24k gold is very bright yellow color. It is also quite soft. In jewelry, pure gold is mixed with other metals to create an alloy, which adds overall strength and changes the color of the metal.
History
The idea that bigger is better when it comes to precious stones was around long before the dawn of precise instruments for measurement. So, hundreds of years ago, when people needed some standard for weighing their gems, they turned to the carob tree. Since the carob's seed weighs approximately the same as the smallest gemstone, it was designated as the base unit for weight, and since the seed was called a carat, the name came too: any stone that approximated the weight of one seed was deemed to weigh one carat.
Carat vs Karat
The name carat derives, via Medieval French and Italian, from Arabic qīrāṭ, meaning "bean pod," which is itself from Greek keration, referring to both the carob bean and a small weight. Early in the 20th century, the weight of the carat was set at 200 milligrams, or 0.2 grams. Thus, a 2-carat diamond weighs 400 milligrams or 0.4 grams.
The word karat is ultimately from the same source as carat but came to refer to the fineness of gold: one karat is equal to 1/24th part of pure gold in an alloy. The 1/24th proportion goes back to ancient Roman times during which a siliqua was equal to 1/24th of a golden solidus. In other words, a 14-karat gold ring consists of 14 parts gold and 10 parts of some other metal, like copper; pure gold is 24 karats.
The substitution of karat for carat in regard to precious stones is considered incorrect, whereas the reverse—using carat in place of karat to indicate the pureness or fineness of gold—is considered acceptable. (English strikes again.)
This use of carat as a variant spelling of karat in the assessment of gold goes back to the 16th century, so its acceptance in contemporary English is understandable.
Today, we commonly encounter both carat and karat in reference to fine jewelry, where size and quality are emphasized, with carat usually designating the size of a stone and karat designating the proportion of gold in an alloy.
It should also be noted that karat is often abbreviated K, as in "an 18K gold ring," but carat is almost always written in full.
Source: Merriam-Webster