The history of the concept of the assault rifle started in the early 1910's, when the famous Russian armorer, col. Fedorov designed a small-bore selective-fire rifle with detachable box magazine. Initially, Fedorow designed a brand new small-caliber 6.5mm cartridge for his rifle, but, due to WW1, switched to the Japanese 6.5mm Arisaka load, which was less powerful than the Russian 7.62x54R and available in quantity. This rifle was aquired by the Russian army in small numbers in 1916 and served (in very limited quantities though) with the Russian and Soviet (Red) Army up to 1925. While the design of the selective-fire rifle was not unique for that time, the concept of the "lightened" cartridge, more suitable for full-auto fire, was new. Also, col.Fedorov invented the idea of infantry weapons families (assault rifle, light machinegun, medium machinegun, vehicle and/or aircraft mounted MGs) based on the same actions and receivers.
The next step in this history was made by Germany - in the 1930's, theybegan research to develop a medium-power cartridge, which would be much lighter than 7.92mm German and easier to fire accurately in full-auto mode. This development led to the 7.92x33mm cartridge (Pistolenpatrone 7.92mm). The Germans developed some weapons designs for this load, including the MP43 and Stg.44, but this was too late for Germany... Further development of such designs was made by German engineers in Spain, and later in West Germany, and led to the HK G3/G41 family of battle&assault rifles.
The United States also put in some effort to this idea, and before WW2 developed a special less-than-medium powered cartridge .30Carbine and a rifle for this cartridge - a so-called "baby-Garand" in semi-auto M1 and selective-fire M2.
But the largest stride forward was made by the USSR, when, in 1943, the Soviet Army adopted a new cartridge - the 7.62x39mm medium-power load. In 1945 , the Soviet Army adopted the semi-auto SKS rifle in this chambering, and, in 1947 - the AK (known for the West as AK-47).