Deep Battle was a doctrine developed by Marshall Tukhachevsky in the 1930s. Tukhachevsky fell victim to Stalin's purges but elements of the doctrine lived on in late war Soviet offensives. Stalin backed this doctrine and it meant that large tank corps were formed in the Soviet union.

The actual doctrine of deep battle essentially consists of achieving as deep a penetration as possibleusing modern weaponry to neutralize simultaneously all of the opposing defences. This penetration was envisioned of going up to 100 kilometres. By 1943 the Red Army were essentially using these techniques with armoured corps attached to local combined arms armies achieving initial encirclements and breakthroughs which the tank armies, held in reserve, extended.

In terms of "countering" Blitzkrieg tactics I'm afraid this is somewhat of a misconception. Traditional German "Blitzkrieg" was over by the end of 1942 - which is when the Red Army began the drive back. The last MAJOR German offensive in the East was Kursk and this was an offensive for local operational gain not the strategic goals of 41 and 42. Kursk was also definitely NOT blitzkrieg as it essentially consisted of a grinding battle of attrition into the teeth of fixed Soviet defences. The beginnings of Soviet Deep Battle in the Second World War essentially began with Stalingrad- although they basically didn't get it right until the Second half of 1943.

Further, Deep Battle is an offensive technique. What you need to stop blitzkrieg is a defensive technique. So while deep battle techniques were used to defeat the German army they weren't ever really a "counter" to Blitzkrieg techniques (Also Blitzkrieg as an operational concept didn't exist as such in the minds of the Wehrmacht but is more of a post-war construction)