The name Molotov Cocktail was awarded during the Soviet-Finnish War (aka. the Winter War) of 1939-1940 to the hand launched incendiary weapons that the Finns were using to destroy Soviet armoured fighting vehicles. These weapons had been used before - certainly in the Spanish Civil War, and possibly as early as the Polish-Bolshevik War in 1920 - but no previous name had stuck. The name was an ironical 'shout out' to the Molotov Breadbaskets - incendiary cluster bombs that the Soviet air force was using to destory Finnish villages. The Soviet foreign minister Vyacheslav Mikhailovich Molotov claimed that the Russian pilots were in fact dropping food to the Finns, and in response they provided Soviet ground troops with a drink to go with the food.
Effectively this is a primitive incendiary grenade. The connection of the weapon itself to the development of mechanised warfare is fairly logical - it consists, in its crudest form, of a glass bottle full of gasoline with a rag tied around the top. The user lights the rag and throws the bottle at the target, it shatters on impact and the burning rag ignites the contents, hopefully setting fire to the target. Variations in construction and use are common - the simplest is to place some form of gelling agent in the gasoline to turn it into a form of napalm - strips of inner tube, liquid soap and foam peanuts are common choices. Commercially made 'cocktails' - as issued to the Finnish Infantry during the Winter War have also been known to include phosphorous so that they can auto ignite on impact without needing a burning fuse. Assuming that the device is not self igniting, the user can also fire one or more 'wet' shots (unlit cocktails) at a target to thoroughly saturate it with filler before following up with a 'dry' (lit) shot to ignite the whole lot.
The Molotov cocktail remained popular throughout the war as an anti tank weapon amongst the infantry of most nations - some of whom actually had them mass produced. The British Army developed a weapon called the Northover Projector - essentially a large spring gun that fired an upsized molotov cocktail, but it was eventually considered too unwieldy and dangerous to the user for widespread deployment.
In the post war period the molotov cocktail became less prominent as a military weapon - armies were better equipped with anti tank weapons and less desperate - and improvements in vehicle design made many military vehicles less vulnerable, but NATO manuals were including instructions for improvising devices in the 25 litre range as late as the 1980s.
Outside the military sphere the cocktail has, if anything, become more popular since the end of the war as gasoline becomes more available - it can be used very effectively for arson and is frequently used against riot police shield walls or for deterring attempts to storm buildings. Since it requires a hard surface to burst on it is less effective against soft targets like hostile crowds, but its use should not be discounted. As an added bonus most civilian vehicles (including police vehicles) lack the engine protection that reduces the vulnerability of military models - also, modern plastic fuel tanks are more vulnerable to molotov attack than the old fashioned welded steel ones.