SOG is the CIA's own, special army. Because the Central Intelligence Agency is a part of the Executive Branch, it isn't subject to the War Powers Act. This is the act, passed in 1973, that limits the President's ability to send the U.S. military abroad; he must notify Congress within 48 hours of deployment, and they can remain in the field for 60 days before he must get the approval of Congress. This was in direct response to our involvement in Vietnam, in which we found ourselves in a very long, costly war, without any approval from anyone. Like I said, the CIA isn't bound by this, because it's not the military. It's intelligence.
Thus, for example, the President could order the CIA to deploy SOG forces to Venezuela to oust Chavez from power, and Congress really wouldn't have a say. Now, of course, that's not how we do that, because then we'd have to send in conventional military forces to stabilize the situation. Also, the CIA is specifically barred from assassination. That's not a mission for the Special Operations Group. No. SOG would go in and destabilize the region, either by causing internal trouble for Chavez (like blowing up electrical transmission lines and cutting phone lines), or instigating trouble with a neighbor (Colombia would be good).
The CIA usually takes people from various military branches and assigns them to SOG. This became prevalent during the Viet Nam war, when the U.S. government wanted soldiers to do something that, perhaps, they shouldn't. Like, say, invade Cambodia or Laos. Or they were assigned missions that required a high level of secrecy. Really, the movie Apocalypse Now gives a good idea of the kinds of missions SOG might go on (not so much with the assassinating Army Colonels, more with the whole infiltrate and kill). The other good movie that gives you an idea of what SOG does would be Clear and Present Danger, wherein the CIA sends down a few soldiers to Columbia to blow up cocaine labs.
These military cut-outs (meaning soldiers who were "cut out" of the normal chain of command) generally come from special forces (though not always). In Vietnam, they might be Green Berets or Long Range Recon Patrol. Today, they would be Rangers, SEALs, or Delta. Marines are also very popular because of their high sense of esprit de corps. In addition to taking military cut-outs, the CIA also trains people for SOG. The training is basically the same as, well, basic training. Only the CIA directs it. Which makes it extra special nasty.
So what does SOG do today? Let's say you've got four forward operating bases (FOB) along the Afghani-Pakistan border. Now, one of these places is being used by a "certain governmental organization," for the questioning of mid- and high-level terrorists. You can't have Billy Bob from the Texas National Guard guarding this FOB, because maybe he owes $20,000 on his truck, and he likes to cheat on his wife, and smokes a little pot from time to time. He's susceptible to bribery by your friendly neighborhood Taliban (money, girls, and drugs). He might be convinced to look the other way when they come to rescue their friends. No no. You get SOG to guard that FOB. Because they're CIA anyway.
That, in a nutshell, is what SOG is, and what it does. Neat, huh?