The weapon later known as Molotov Cocktail appears to have had its operational debut during the Spanish Civil War of 1936-39, when it was first used against Nationalist tanks by the Republican forces. From these beginnings this weapon soon became used by most nations as it was an easy weapon to produce and use, both by regular forces and irregular forces.
During the Winter War (1939-40), the badly-equipped and heavily-outnumbered Finnish Army faced the powerful Red Army after the Finns refusal in surrender some land to the Soviet Union. So they used the improvised incendiary device against the Soviet tanks.
It is during this conflict when the name for this device came from. It was derived from the Soviet politician Vyacheslav Mikhailovich Molotov due to his radio broadcast telling that the Soviet Union was not dropping bombs but rather delivering food to the starving Finns. As a response, the Finns called the incendiary bombs Molotov picnic baskets, later baptized as the Molotov Cocktail. In the beginning, the term was used to describe only the burning mixture itself, but with time the term started to be applied to the combination of both the bottle and its contents.
The Molotov Cocktail was used in the Winter War, then in the Continuation War (Finns against Soviets) and used by all sides in World War II, mainly due to its quality of damage the enemy morale. For instance, during the Warsaw Ghetto Uprising in 1943 the Molotov Cocktail was used against the Nazi troops:
The well-aimed bottles hit the tank. The flames spread quickly. The blast of the explosion is heard. The machine stands motionless. The crew is burned alive. The other two tanks turn around and withdraw. The Germans who took cover behind them withdraw in panic. We take leave of them with a few well-aimed shots and grenades.
The basic weapon is simply a glass bottle containing petrol (or some other inflammable substance) with an oil soaked rag or something similar around the neck. This rag is ignited immediately before the weapon is thrown at a target, the breaking of the bottle as it hits its target allowing the contents to be ignited.
The main way to ensure damage is to detonate the bottle bomb over or near the engine louvers or perhaps the vision devices. It was also discovered early on that petrol alone was not a very efficient anti-armour weapon as it simply runs off the sides of a tank even as it was burning. In order to make the flame-producing mixture stick, the petrol had to be mixed with a thickening agent such as diesel or oil or in some cases various forms of latex.
Late in the war, the Soviets attached bushes or wire mesh to protect the rear end of the tank, hoping that the bottle wouldn't break, as it wouldn't hit the armour, but the Finnish solution was to tie 2 or 3 stones at the end of strings and tying the strings on the bottle so that the stones would shatter the glass. Also barbwire was wrapped around the bottle, so that if the bottle hit the mesh protecting the tank ventilation, the chance of setting the engine on fire increased.
There were several of these types of grenade but typical was the British Grenade, self-igniting, phosphorus, no. 76. This was a glass milk bottle with a pressed-on cap (containing a mixture of phosphorus, water and benzine) and was intended primarily for the anti-tank role. It could be thrown at its target or launched from the Northover Projector, and contained a piece of smoked rubber that gradually dissolved in the mixture to make it stick better to its target. Each no. 76 grenade weighed about 0.535 kg.
Even today, the Molotov cocktail it's a common weapon of terrorists and rioters.