The Spetsnaz were organized more like massive use of SAS raiding teams. A Spetsnaz brigade of 1300 men could field about 100 8-10 man teams. A Spetsnaz company had 135 men, further divided into 15 independent teams. The actual organization of these brigades was four parachute battalions, an assassin company, a headquarters and support troops (mainly communications). A naval Spetsnaz brigade had two battalions of "combat swimmers" (comparable to U.S. SEALs), a parachute battalion, a midget submarine company and other units the army Spetsnaz brigades had. There were also many independent Spetsnaz companies assigned to armies or smaller units.
In wartime, each team would be given an objective to destroy deep inside enemy territory. Or, if not to destroy something, to go deep and find out what was happening in the enemy rear. Put simply, the job of the Spetsnaz was reconnaissance and sabotage. The Spetsnaz teams would get to the target by parachute, ship, submarine or as "tourists" before the war began. At the height of the Cold War, the Soviet Union had about 30,000 Spetsnaz in service.
There was one flaw with this system; most of the Spetsnaz troopers were conscripts, in the army for two years. The Russians made this work by being selective in who they chose to be in the Spetsnaz, and putting the recruits through a rigorous, and violent, training program. You could think of the Soviet era Spetsnaz as paratroopers with additional training in demolitions, infiltration techniques, foreign language training (which many Russian conscripts had just received in high school) and reconnaissance techniques. Perhaps most importantly, the Spetsnaz recruits were taught to think for themselves. This was a rare directive in the Soviet (or Russian) armed forces. But for commandos to be effective, they had to think independently, and the Soviets realized this when they set up the Spetsnaz and the Spetsnaz training program.
During the Soviet period, the Spetsnaz were seen as an elite organization and a career enhancing thing to have on one's resume. The army had more volunteers than it needed and would take the top graduates from the training program. A favorite method was to send volunteers to the six month NCO course. This course had a high wash out rate, but those who made it through were competent leaders and just the kind of people the Spetsnaz were looking for. Even after the Soviet Union fell, the Spetsnaz were still seen as an elite. It did not go unnoticed that Spetsnaz veterans were always in demand as well paid bodyguards and security experts.
The Soviets knew they were getting a lot of eager, motivated and not thoroughly trained Spetsnaz troopers. But they had so many of them that it was felt enough of them would do enough damage to make it all worth while. We'll never know if the original plan would have worked, but the Spetsnaz were effective during the 1980s Afghanistan war. The main reason wasn't the superior Spetsnaz combat skills, but their initiative and ability to think for themselves. The Afghans they were fighting noted this, and learned to clear out of the area if Spetsnaz were found to be operating there.
The Spetsnaz recognized the need for career troops for some jobs. The assassin company in each Spetsnaz brigade was staffed with 70-80 career soldiers, whose job was to find, identify and kill key enemy political and military leaders.
When the Soviet Union fell in 1991, the Spetsnaz didn't disappear. The new nations formed from parts of the Soviet Union inherited any Spetsnaz units stationed in their territory. Many of these non-Russian Spetsnaz still exist, although most are not of the same quality as they were when the Soviet Union still existed. Although there are fewer Spetsnaz today, there are still about 10,000 of them in Russian service. And more of them are career soldiers (more than half, versus 20-30 percent during the Soviet period.) Many of the current Spetsnaz are specialists, with specific skills needed for underwater operations (like U.S. SEALs) and anti-terrorist operations (like the U.S. Delta Force.) But there are still eight brigades (seven army and one navy). The post-Soviet Union Russian government maintained the strength of their commandos because they knew they would need some skilled and dependable troops for emergencies. Unfortunately, the corruption that swept the rest of the armed forces has hit the Spetsnaz as well. Officers stole and took bribes, as did NCOs, while the lower ranking troops developed a bad attitude towards military service. During the 1990s, the government raised the pay, and tried to improve the living conditions for the Spetsnaz, lest they lose the loyalty of the most effective troops available.
After World War II, it took the Soviet Union a while to note the success of U.S. and American commandos and attempt to emulate their success. In the 1960s, the Red Army began to organize "troops of special purpose" ("Spetsialnoye nazranie", or Spetsnaz for short) units. The Soviet Union had always had some form of commandos, but there were special units of the secret police (KGB.) For special operations, the army would form temporary units consisting entirely of officers.