All the Basics:
Neti-Neti meditation is a form of self-inquiry in which the practitioner search for the location of the self (Neti-Neti Meditation). The name, from Sanskrit origins, translates to "not this, not that," which means that he is not limited to his thoughts, body, or ego.
Through this form of meditation, the practitioner realizes that he is not tied to the things he normally identifies with, whether it be personality, thoughts, feelings, emotions, memories, sensations, bodies, perceptions, will, gender, sexual orientation, or jobs.
How to Perform the Meditation:
Sit in a comfortable seated position. Close your eyes. Begin to make a list of what you are not. For instance, "I am not this wall. I am not my pants. I am not my job." After you run out of things that you are not, you can begin to realize your true identity and start to think, "I am..."
Mantra, Sound, or Visualization:
For this meditation, visualize what you are not, before moving on to true self-discovery.
Origin + History:
This form of meditation is a Hindu practice, found in the Upanishads and the Avadhuta Gita and constitutes an analytical meditation that helps the person understand the nature of the Brahman by first understanding what is not Brahman.
It's a form of Vedic meditation, and Adi Shankara was one of the foremost Advaita philosophers advocating for this approach. In his commentary on Gaudapada's Karika, he explains that Brahman is free from adjuncts and the function of the practice is to remove the obstructions produced by ignorance. His disciple, Sureshvara, explains that the negation does not have negation as its purpose, but rather is more concerned with identity.