(pictures will be added to this post later as time allows)
There are a LOT of ways to grip a handgun, and each is appropriate to different situations, different hands, and different guns. There are even a lot of very small variations to specific grips.
For decades, the 1911 was shot with the bullseye grip (and the bullseye stance) usually with the tucked under thumb. This position isn't very fast, although it can be extremely accurate for longer distance shooting. The marine corp was teaching this style of shooting all the way up until the adoption of the M9.
The thing is, the bullseye grip was developed for; and is best used with, revolvers. It is designed so that it is easy to re-cock a single action gun with your dominant hand, as well as be useful for a long double action trigger pull (and is especially good with a long barrel). The problem is, the bullseye is a poor hand position for using the safety, and poor for rapid fire and rapid re-acquisition I.E. it's bad for recoil control. Also, it is just bio-mechanically "not quite right" for the shape, and balance of auto pistols.
Since the '50s and especially since the development of IPSC; there have been a lot more grip styles tried, refined, rejected, folded, spindled and mutilated.
What most competitors have settled on for their (mainly) 1911 style pistols is some variant of what is called the "Gas Pedal" grip; where the dominant thumb rests on, and pushes, the safety like a gas pedal. In fact, there are specific safeties developed for the 1911, which look like gas pedals; which is where the style gets its name.
Of course, as I mentioned above, even within the gas pedal grip, there are near infinite variations, from the "hitchhiker", to the "tucked, full high wrap, thumb over thumb".
The classic gas pedal grip has the dominant thumb folded over the safety, with the support hand wrapped round the bottom corner of the gun and your dominant hand, in what's called a "half teacup"; and the support thumb pressed either into the top of the dominant hands middle finger; or into the support side of the trigger guard.
The gas pedal is fast, natural and easy to index into, points well and gives good recoil control. Unfortunately, it's very easy to accidentally wipe off the safety when you didn't mean to; it can be somewhat inconsistent, inducing twisting into your grip; and you can easily accidentally hit the mag release (especially on modified race guns).
I used to use the classic gas pedal grip, but I found that the safety could snag on things, or get wiped off at the most inappropriate times, and that my presentation was slower, less natural, and less consistent than I wanted.
I switched to a variant of the thumb over thumb gas pedal grip a while a go, and while it's not quite as naturally easy to control recoil, the recoil control is actually stronger (it just feels weirder), and I get good results.
With the thumb over thumb, I'm personally faster and more consistent in presentation; and my thumb positioning gives more positive control over the safety.
Now, as I said, I use a variant of the thumb over thumb: the high, full wrap, forward, thumb over thumb, either tucked or untucked depending on the situation.
In this grip, the gun is drawn with the second knuckle of the support hand thumb tucked under the safety and indexed forward, with the safety on, and the pad of the dominant thumb pressed against the second knuckle of the support thumb.
For non-competition draw, the tucked version of this grip is used; and the dominant thumb is kept tucked under the safety; until when ready (presenting the gun), the thumb is brought over the top of the safety, wipes it off, and THEN presses firmly on it (negative pressure), to assist in recoil control.
Importantly, until the weapon is presented, positive pressure is kept on the dominant thumb; keeping the safety in the up position until it is to be wiped off.
Whether tucked or untucked, some folks also keep negative pressure on the slide stop to prevent the stop from being accidentally engaged during recoil, but I think that's a bad idea. I like to index the tip or side (depending on high or low wrap) of my thumb on the pivot pin of the slide stop.
In competition draws, or whenever the weapon will be presented immediately, the thumb is always kept on top of the safety for speed and consistency; and the support thumb knuckle and muscle at the heel of the hand are used to keep positive pressure on the dominant thumb; keeping it from wiping the safety off accidentally.
In this google video, you can see Todd Jarrett demonstrating the LOW wrap version of the forward thumb over thumb grip
I'm almost entirely in agreement with Jarret, I'm a BIG advocate of the full wrap grip; I just use the high wrap, instead of the low wrap.
What's the difference?
In the high wrap, I index both thumbs forward; my dominant index finger along the frame and top of the trigger guard (presuming a 1911), and my support hands finger wrapped around the trigger guard past the distal joint, and into the medial joint if I can (some guns that doesn't work for). This is also called the "Plaxco Grip", after James "Mike" Plaxco (an IPSC shooter) started using it some time back. Of course theres no coincidence, that Mike has HUGE hands; and he also likes REALLY skinny 1911s (he "invented" the thinned competition 1911 grip frame in the early 80s).
In the LOW wrap, all of the support hand fingers are wrapped UNDER the trigger guard, with the support pinky finger wrapped under the dominant finger, giving a very slight teacup. This is also called the "Enos", "Leatham", or "Enos-Leatham" grip after Rob Leatham and Brian Enos, two well known IPSC shooters who started using it in the late 80s; and have
there's not a lot of difference between the high, and low wrap; really its a matter of comfort, preference, and hand size.
If you have large, strong hands, the high wrap is very effective at reducing muzzle flip. If your hands are smaller, it isn't very effective; but wrapping under the pinky with the low wrap can be, especially is you use dynamically opposed tension in your grip technique (one hand is pushing forward and down, the other pulling up and back).
There is one other big difference between the high and low wrap (and the classic gas pedal); and it has to do with the shape of your safety. With the classic gas pedal, and the low wrap position, a long, wide gas pedal safety is much better. It's faster, and gives more support to the dominant thumb.
In the high wrap position, your dominant thumb is supported by the knuckle and heel of your support hand; and because they are tucked in so close, the wide gas pedal gets in the way; slowing down your safety engagement and disengagement, and potentially wiping the safety on under recoil - which is a BAD thing.
For years, I've used the gas pedal; but I don't like how thick it is. It snags on things, and catches on my thumbs; and it really just isn't necessary for anything other than speed shooting. Just yesterday I switched from an ambi gas pedal, to a slimline extended single side. I like the way it handles a lot more; and I'm sure it's going to clear and present better. I don't have to move my thumb as much, and the narrower extended rib doesn't drag on the heel of my hand from the high grip.
Oh and there is one other thing I disagree with Jarret on, and that's the square isoceles stance. For speed shooters the square stance isoceles position is great; but for combat shooting, I think it's a bad idea (for MANY reasons).
I personally use a modified weaver, that is a combination of a fighters stance, weaver, and isoceles. It's fast, presents a smaller target area; is well balanced and maneuverable, and is easy to present to. No, its not as fast or easy to present to as the square isoceles, but I think it works better in the real world rather than at an IPSC shoot (especially from partial cover).