Okay, let me start by saying that there are at least three widely recognized "meanings" for NLP:

In connection with Transcendental Meditation, NLP refers to the "Natural Law Party".

In the world of computers, and AI (artificial intelligence), etc., NLP stands for "Natural Language Parsing/Processing".

In the area of psychology - and on this website - NLP stands for Neuro-Linguistic Programming, and if you think that's a bit of a mouthful you may well be right.  Still, it is a pretty accurate description of what this field of study is about:

Neuro

The activity throughout your brain and your whole nervous system - why what you think affects what your body does and vice versa.
 

Linguistic

Refers to words - and how we use them: how we are affected by the words we use, how our perceptions are framed by the words we choose.
 

Programming

Taken from the computing term, this basically means that as we go through life we collect all kinds of "programs" (similar to what are referred to in Transactional Analysis as "scripts.")
The key idea is that just as a computer will try to execute all of the instructions in a program, regardless of whether they make sense, so human beings tend to act in accordance with the ways in which their experiences have "programmed" them, even when the program doesn't make sense (like people speaking very slowly and loudly to foreigners as though that will somehow improve the level of communication).
On the positive side, just as computer programs can be debugged, or even completely re-written, so we can examine the mental programs we live by and, if we don't like the results they generate, we can "re-write" them to get results we do want, or we can adopt completely new programs that have been shown to be effective for someone else.  In other words, NLP is about our ability to change our lives in the way we want by changing our thinking and our behaviour.

Important Note:

According to Korzybski, who first coined the label, there should always be a hyphen in "neuro-linguistic" to indicate that this refers to two individual entities - neurology and linguistics - working together, as compared with "neuro linguistic" (two separate entities) or "neurolinguistic" (a single entity), neither of which make this distinction clear.