Neuro-Linguistic Programming is the study of how people think, remember experiences and learn.
Every time someone is speaking they are giving away how they think by the words they use, the way they breath, the non verbal movements in their body, their muscle tension, their eye movements etc.
NLP is a set of tools and skills that allows you to understand this in a way that allows you to be in rapport with the other person. To intervene at a level of core transformation, to influence, to sell and also to learn what the other person who may be a genious is able to do.
It also teaches you what to look for specifically. What to listen to? There are millions of things happening around you right now to which you do not pay attention. There is a specific way your feet feels, the difference in lighting in different parts of the room, the warmth of your clothes on your body, the distinct sounds that you hear. The rhythm of your heart beating and the way you breath and how it makes you feel. There is a reason why you are not aware of everything at the same time. Because that will be overwhelming. NLP points you precisely in the direction your focus of attention is useful.
It is primarily a tool of influence that you can use by transforming yourself to pay attention to the way the other person thinks. And it is so obvious and elusive that when you learn even just the fundamentals you may wonder how could have you been blind and deaf to these obvious patterns happening in front of you all the time.