The pistol formation was invented by Nevada's coach, Chris Ault, around 2003/2004. Basically, it combines the shotgun spread attack with the read option. The basic formation has the QB in shotgun with a RB behind the QB. However, in the Pistol the QB is roughly 3 or 4 yards behind the center, unlike the normal shotgun where he lines up of roughly 7 yards. As a result, the RB is roughly 3 yards behind the QB, and thus 7 yards behind the center. This creates a few unique advantages. First, the QB is closer to the line of scrimmage, and this allows him to have a better and quicker read of the defense. Also, the ball is snapped to the QB much quicker in the Pistol compared to the Shotgun because he is closer to the LOS. Next, the QB has two options, pass or run. If he chooses to run the ball, then he uses the "Read Option" and normally will read what the action of the weak side defense end/sometimes OLD depending on if the defense is 4-3 or 3-4 (The weak side is the opposite side of the tight end on the O-line). Under the basic premise, the RB runs between the tackle and guard, but if the QB keeps the ball he runs outside the tackle.
Lets assume the defense is a basic 4-3 with the tight end on the right side of the line, and the ball is snapped. Both the QB and RB start to run to the left side of the line. On the O-line, the center will take out the DT, the guard takes out the middle LB, and the tackle will either take out the DE or weak side LB depending on who is closer and what action the DE takes. Let us assume the DE crashes in, and tackles the RB, then the QB takes the ball and runs outside the tackle. On the other hand, if the DE plays outside, and contains the QB, then a lane is opened up between the OT and OG, and the RB runs between them.
What makes the Pistol deadly is how it can setup the play action pass. Again, in its basic form, you have 3 WR and 1 RB. Assume that it is the 2nd Qtr and during the 1st Qtr you ran the Read Option on 1st down every time. However, this time you decide to fake the run, and pass. As a result, the LBs first response will be to run to the LOS to stop the run, and the middle of the field/behind the LBs will be wide open. Therefore, in a 3 WR set with a TE, you can send either the TE or 1 of the WR in the middle of the field, and he will be wide open. Again, this is just its basic form. In a nutshell, this is what the 49ers did to the Falcons last week.
In the NFL and even in college, they are using different formations with the Pistol to confuse the defense. For example, the 49ers sometimes place 2 RBs on both sides of the QB, and this confuses the defense because they dont know which side the "Read Option" will go to. Also, they even had 3 RBs at times, two on either side of the QB and one behind him. However, a lot of the formations are based on the personal that can be used.
The Wildcat is when a Non-Qb, normally a RB or WR (most of the time they played QB in high school) lines up in shotgun and takes a direct snap. He can either run the ball, or pass it. In theory, this allows the offense an extra blocker because you do not have to account for the QB anymore because you eliminated him as the middleman, and got the ball to the playmaker right away. However, the main problem with this is that the RB or WR is not that great of a passer, and the defense will stack the box with 7 or even 8 guys thus making it very difficult to run the ball. Even though there is an extra blocker, for the Wild Cat to be successful you really need the person receiving the snap to be a real passing threat.
Now, the counter is that the QB can not afford to take that many hits, aka RGIII. The latter is correct in theory, but IMO for the Pistol to work in the NFL all you need is the threat of the QB to run. For example, before RGIII got hurt against the Ravens he was running it roughly 8 to 10 times a game and averaged 6 yards a carry. On the other hand, Kapernack only ran it 5 to 6 times a game and averaged 6 yards a carry. Furthermore, in the Atlanta game he only ran the ball twice. Just the threat of him running the ball had the Falcons move their DE more towards the outside, and this opened up running lanes between the tackle and guard and allowed Gore to average 4 yards a carry.
IMO, the Wildcat was a fad and defenses were able to quickly adjust to it. However, I believe the Pistol is legit assuming you have the right personal to run it, starting with the QB. The reason it is dangerous, and different than the Wildcat, is due to a legit passing threat. Therefore, the defense has to respect the passing game, and can not stack the box. As a result, this opens up the running game with the "Read Option". Not only does the QB have to be a running threat, a passing threat, but the most important part is that he has to understand how to read the defense. If the QB is unable to read the defense, then he can have all the physical skills in the world, but it wont matter. The best example for he ideal QB is Kapernack because not only does he have the physical tools, but he learned it in college under its inventor. Also, towards the end of the season, the Seahawks started running the Pistol with Russell Wilson. An example of a QB that has the physical tools, but not the intelligence is Cam Newton.
Finally, someone was talking about the West Coast Offense. In theory, a "Pure/True West Coast offense is based on the following. Use the running game to open up vertical passing lanes. Then use TEs and RBs to setup a horizontal passing game that stretches the field, and thus keeps the defense off balance. Also, the timing of the passing game is essential in the West Coast, and is based off of either 3, 5, or 7 step drop backs. The West Coast offense requires a team to have a TE that can catch the ball, and a RB that can catch the ball in the flat.