The centerline is a very important concept to striking. It serves many offensive and defensive purposes— understanding it will vastly improve your overall efficiency and effectiveness.
Luis Monda (known rather well as Sinister on Sherdog) is who he's referring to in the "excellent thread". Every image and everything in quotation is from his thread here.
"Theoretically, when two fighters square off they're facing each other... I like the lead toe to be aimed at the opponent's centerline, as well as the lead hand, hip and shoulder. This gives them a direct threat. Early boxers did this without deviation, and regardless of the proximity of the opponent (how close they were)."
"Control of distance is the first principal of boxing defence, followed by positioning. Your stance can control distance by having depth, and appear threatening. You LOOK as if you're prepared to hurt anyone who steps close to you. Positioning is never sacrificing your stance, or that threat... if it can be avoided. If you're almost always facing the opponent, people will find it very very difficult to initiate attacks."
Simply put, when you line up and aim your choice of attack (linear in this case) at the centerline, it becomes more efficient and effective. If your choice of attack is the jab, then lining up your lead hand with the centerline of your opponent does several things.
Since the lead hand is already aimed at the target, all that needs to happen is to be fired down the line. This means no further adjustments need to be made in order to fire the punch (unless your opponent adjusts off the centerline of course). This concept is similar to having a target in your scope vs. readjusting your scope. Lack of readjustment means faster strike and often more effective strike.
Stance-wise, most boxers today keep their head in the centerline, but keeping it off the centerline is not a lost art. You need only look to Mayweather, Bernard Hopkins, and even recently Timothy Bradley (vs Marquez) to spot it. Old timers like Joe Louis, Sugar Ray Robinson, and Charley Burley also had a profound understanding of this theory (built into their stance), and they were truly brilliant.
Though this youtube video is describing more of street fight situations, it does an excellent job at introducing the centerline concept.
The excerpt below is from Loren W. Christensen, Wim Demeere in their book: Timing in the Fighting Arts
This is some advanced explanation and application of the centerline concept. In my opinion, it is done by a genius and discusses a brilliant fighter: Charley Burley.
Notes from and for the video:
"In no sport is the temptation stronger than in boxing to run before you can walk. It is a temptation more strenuously than some others to be resisted. You must learn to stand in position before you begin to hit..." J.G. Bohun Lynch : sound familiar?
1. You should always stay cognizant of the centerline and it should always be protected, whether reactively or proactively (built-in defence).
2. Your own centerline is always considered in relation to the opponent's centerline. This consideration dictates the arsenal of attacks and the angles in which they can be delivered.
3. Footwork and rotation are the subtle motions that determine the employable attacks, and you must readjust to react accordingly.
4. Being able to create a line of attack while your opponent has to readjust puts you at a very significant advantage. This can often mean creating a strike by going into an "inside-angle" or an "outside" angle. Other options include creating a counter with evasion or deflection. *All of this will be discussed in the future.
5. The centerline for punches can be protected by holding a high guard, which blocking the path of the centerline. Other defenes as can be employed by deflecting the attack via a parry, shoulder roll, or evaded via a slip, lean, or duck. Remember that in a street fight / MMA fight, the gloves aren't there as a barrier.
6. By always facing your opponent (regardless of all the movement happening), you are ensuring that you're staying cognizant of the centerline.
The centerline and line of attack is so very important to striking because it gets you to think about the subtleties that matter. Striking is truly a game of chess in the sense that whoever is more ready (in position) to launch a strike will most likely be more effective. And although it's a concept that is primarily for linear attacks, understanding it will set up arced attacks as well.
Pointing your lead hand and foot at your opponent is just one illustration of using the centerline to your advantage. Other adjustments can be make accordingly for your choice of attack. For example, if your choice of attack is a cross, you can bait your opponent in while adjusting your hips and foot to suit the rear hand strike: e.g. Marquez vs Pacquaio
For an illustration of adjusting linear attacks to the centerline, refer to Junior Dos Santos vs Mark Hunt. You will notice that JDS plays with the concept quite a bit, he attacks the centerline with the lead and the rear (subtle adjustments of his positioning for both).
You'll see that he points the jab at Hunt quite often and then quickly readjusts to sneak in other attacks. The constant threat of centerline (both body and head) then allows him to go for the big overhand right as the opponent possibly thinks its a bodyshot.
The end of the fight also plays with the threat of the centerline: because JDS has established his jab so well throughout the fight, Mark Hunt opts to parry it with the rear hand (common defence). Though Hunt returned the hand immediately by sensing something is wrong, he did an outside slip which caused him to go towards the kick. This allowed an opening and a powerful blow for JDS' spinning back heel kick to finish the bout.