Personal Reality

Although nothing has been disclosed on the true meaning of Personal Realities, it has been repeatedly stated it is the source of an Espers abilities.

Function
Apparently, all human beings have a Personal Reality within their minds; specifically their consciousness. It is possible to develop it individually, and will lead to that person attaining supernatural powers, therefore making them an Esper. The more powerful or more growth a Personal Reality gets, the stronger the powers of the Esper are.

According to Uiharu Kazari, it is related with "beliefs" so in essence one's esper powers grow if they strongly believe they will. A teacher of her and Saten Ruiko explained that probability is also an important factor.

Personal Reality is also related to Quantum theory. In quantum theory, you don't know something until you actually measure it, for example Schrödinger's cat (you do not know if the cat is alive or dead until you open the box) Komoe Sensei explains this to Index using a chocolate box. She asks Index what is inside the box and she replies chocolate, but when opened it is candy. Espers have attained a state of mind where they are able to project their Personal Reality on to the universe's reality, therefore giving them powers (i.e. they are able to manipulate what is unseen in the box or change the position of magnetic field lines as in Misaka's case)

It also seems that it is the source of an Esper's AIM field.

Development
It seems that there are multiple methods to give the Personal Reality growth. A commonly seen test is the use of Zener Cards where the student cannot see the cards, but must guess the right symbol.

Other methods include using drugs and electric shock to change a person's brain. There are also methods of training abilities; such as 'Columbus' Egg', where a telekinesis esper have to balance a egg upside-down without breaking the egg open (this is considered a high-diffculty technique); and also playing Poker blindfolded, and sometimes as punishment for students they must win 10 times consecutively before allowed to leave the catch-up classes. Also, according to Kihara Amata, the values a person inputs or sets into their Personal Reality can affect the abilities produced, and that each ability has a 'true form'.

It's growth is correlated with the brain's processing ability, and can become more powerful if there is a stronger 'processor', such as a result of Level Upper. If that processing ability is deterred (e.g Capacity Down), the power can either lose its strength or become unusable, such as in Accelerator's case after being shot or Kuroko's inability to teleport when she can't concentrate. As shown by Kiyama Harumi, if one controls another's conscience (and therefore his/her Personal Reality), they can access that person's powers.