Primes are the highest ranks of Cybertronian titles. It is typically used as identification for the leaders. Only gained by being given the matrix of leadership. An object of pure limitless energy and power. In IDW ( comic) the Cybertronians believed the Matrix was a made up thing in order for the primes to keep them in line. It is also in the key to a dangerous weapon. One that can destroy our sun. ( Bayverse)

The matrix chooses who is worth of being prime itself. It will light their darkest hour. Yet there is always an equal or opposite force. It goes by two names Dark Spark and the Anti-Matrix. I have to infiltrate top secret Decepticon data archives for more info on the Dark Spark. But I've heard Lockdown had it last.

   TheMatrix is made from from pure crystal energy. It can even revive Cybertron if it dies. It can make a spark flicker back to life. The Matrix contains all of history. Which is why primes are so wise. ( also there's a secret with the Matrix((IDW)) )

The Allspark is what we call the Afterlife. Where we die we 'become one with the Allspark'. It is a million moving sparks. It is tied directly with Primus. In some universes it is a physical cube that grants us life ( since it ties with Primus) and keeps the planet alive ( basically the heart of the planet.) Since it is a million moving sparks when a transformer is borne or created a spark breaks off and enters the transformer, granting it life and soul. When they die the spark returns to the Allspark with the memories and experiences intact.
This is why in one universe Megatron started the war. To gain the Allspark and reform Cybertron to his fitting. Optimus made the ultimate call to shoot the Allspark deep into space where it would buy the Allspark time to escape Megatron. But by doing this seals the doom of Cybertron. (Since it is the heart of cybertron)

The Well Of Allsparks is a hole that goes to the center of Cybertron. Into the heart of the planet. In Tfp Optimus had to merge the Allspark with Cybertron to revive the planet. (Omg so emotional, Transformers Prime)
Vector Sigma is a supercomputer (more like MEGAcomputer) that grants new transformers with life. For example in G1 the Autobots created the Aerialbots and Vector Sigma granted them life. Or when Megatron created the Stunticons with Vector Sigma. In beast wars one could communicate with the Allspark via vector sigma with the Oracle.

The Matrix of Leadership
   (Alternate universe

   (Our universes)

   (The Allspark)

(The Well of Allsparks)