This is a question I get asked a lot and I must confess, I never answer it the same way twice. NLP is such a vast field that it is difficult to put it into a nice tidy sentence or paragraph.
Perhaps the best way to explain it is to tackle each word on it's own.
Neuro refers to our mind. We experience life through our 5 senses and we combine this information with our state to create meaning. The meaning is subjective because we only take in information that is pertinent to us and our state is always changing.
Linguistic refers to language. The information that comes into our brains is stored through the use of language. Think of it as a filing system where we group our memories together by attaching words. When we have a new experience we tend to make sense of it by finding where it fits in our history of memories. These memories are retrieved based on the language we used when we filed them.
Programming refers to the fact that we interpret information in predictable ways. We filter information in ways that are unique to each of us and this results in patterns of thought.
The vast majority of information we are exposed to must be deleted in order for us to make sense of the world. It simply would not be possible to process all stimulus all the time. We notice things that are pertinent to us and we do so at the expense of all other stimulus. This creates a predictable thought process which in turn leads to predictable behaviour. It is easier for us to recognize predictable patterns of behaviour in other people than it is in ourselves.
It is this pattern of behaviour that exists beneath the level of consciousness that makes an NLP intervention so life changing for many people. When you can see with clarity how you have been doing your problem, your problem tends to dissolve.
This means that NLP is really the language of influence over our own minds. If the meaning we assign to events in our lives is subjective and based almost entirely on our past experiences, we can learn to create meanings that lead to more effective behaviours. It is this notion that there is no inherent meaning in anything and that the meaning of any stimulus is assigned by us that produces different results. Getting our minds around this concept can be a challenge but NLP has many techniques that make it much easier.
Many of my breakthrough clients tell me that they just "see things differently now." I sometimes think about the fact that there new view of the world may not be any more accurate than their old one, however it is definitely getting the results they desired. In my subjective view, that means that their new way of seeing things is better.
You are assigning the meaning, so you may as well assign a meaning that leads you towards your goals.