Solid gold is a term used for various designations of karat gold. The purest form of gold is 24 karat, meaning 24 out of 24 parts are 99.999% gold.
All other karats of gold are technically gold alloys, meaning it's a mixture of gold and other metals. For example, 14k gold is 14 parts pure gold and 10 parts other metals, like silver, copper, nickel, etc.
24k is very expensive and very soft, so other metals are added to make the gold more affordable and durable. 14k and 18k are most commonly used in jewelry production for those reasons.
The exact formulation varies depending on the desired karat and color (yellow, rose, white).
Gold vermeil refers to sterling silver (.925) or fine silver (.999) electroplated with karat gold.
The industry-set guidelines require that (1) the base metal is silver, (2) at least 10k gold is used for plating, and (3) the plating thickness be at least 2.5 microns.
Some gold vermeil jewelry has a ".925" or ".999" mark to denote the base metal is silver.
Gold filled is created by mechanically bonding karat gold and a base metal. The name can be misleading but think of it as a tube of gold filled with another metal, typically brass.
To qualify as gold filled, at least 5% gold by weight must be present. Often gold filled metals are stamped with a "1/20 14k" mark indicating 5 percent 14k gold.