Most people associate this in meditation with the monkey brain, babbler, mind chatter etc. It is when you are attempting to meditate and remain silent within your mind, and thoughts of doing laundry, paying bills and an assortment of other things intrude your mind distracting you from your goal of non-thought. All forms of meditation make this a precedence to achieve inner silence. Once inner silence is achieved nirvana is believed to follow. Unfortunately, this is also a very commercialized concept and largely misleading through commercialization of meditation to westerners.
Those who stick it out over days, weeks, months and years and achieve a level of controlling the babbler do find a form of inner peace. They essentially remain in quiet solitude day in and day out. This unfortunately is not what the mystics had in mind for their apprentices. The first mistake is in the word non-thought, it suggests to not babble in one's mind, and this is correct but it does not mean to not have thought or think. This is the difficult part to grasp.
Wherever you are, right now for instance, there are objects around you. Perhaps it's the computer screen, a table, chair, pen, paper speakers etc. You are aware of these objects, and each of these objects has what we call a "feels like". For instance, without touching the monitor you can image the texture of the glass, the temperature of it etc. This is called a "tag" or a "feels like". The desk or table also has a feeling of texture, density and temperature this is also called a "tag". You can imagine all of these things yet there is no need to shout out the word TABLE or MONITOR to define those objects. This is called awareness and it is a form of non-verbal consciousness. So when masters suggested having non-thought they meant to not have verbal words in your head. Of course this is not simple to achieve, but with practice you will get the hang of it. When you understand this concept and can internalize it as a function, you can transcend the boundaries of everyday life. For instance, language is no longer a limitation; you could even say that the universe could communicate with you, if you were ready to listen.