1) Spend time really reading this site (it is in ‘citizen-speak’. You will find much neuroscience written in understandable science).
•Read articles in the BLOG section.
•See videos and links.
•Get in depth understanding in why NeurOPTIMAL is the safest and most up to date brain-biofeedback (neurofeedback) technology in use today.
2) Listen to this interview with Valdeane Brown, PhD., the developer of NeurOPTIMAL neurofeedback for this reason: Here is the link.
Val Brown, Ph.D., is a trainer of trainers and a leader in the field.  He has owned, operated, been certified and taught almost every major neurofeedback system in use today and knows what is happening ‘under the hood’ in all the systems in use.  He is a great authority of where the field is today and how our approach is up to date with current physics and math, and in line with the most recent research and understanding about how adaptable and plastic our brains are, if we know how to mirror back to them their own dysfunctional behavior. Here is the link.

3) This article mentions the American Academy of Pediatrics endorsement.  

4) Research:
4a) Robbin’s Article Wired for Miracles is a good historical overview of the field and how it developed.  He is not up on the latest developments of non-linear, dynamical neurofeedback where we neither diagnose nor drive change with protocols.   But none the less, it will give you a sense of how the field has developed.
4b) Here is the link to a meta-analysis, or what is an analysis of many studies of neurofeedback for ADHD. http://www.ncbi.nlm.nih.gov/pubmed/19715181
BCIA Website for research