What is the difference between 14 karat and 24 karat gold?

The purity of gold is measured in Karats, with 24 Karat representing 100% purity. The karats are defined as

K = 24 (Mp/Mt)

where, Mp is the mass of pure metal and Mt is the mass of total alloy.

Hence 24 Karat gold will have 24 parts of pure metal and 0 parts of other metals, making up the alloy.

14 Karat gold will refer to 14 parts of pure gold and remaining 10 parts (=24 parts-14 parts) other metals. In other words, 14 karat gold is 

(14/24) x 100 % = 58.3% pure.

In comparison, 24 karat gold is 100% pure.

Note that 100% pure gold is very soft and as such can not be used in jewelry and other applications. Addition of other metals provide these properties and as such, people buy 22 karat or 20 karat or 18 karat jewelry.