What is Neuro-Linguistic Programming

That's a surprisingly difficult question to answer concisely.
 I like to think of it as the nearest thing we have to a blueprint of how the unconscious mind works.  One of its creators described it as "An attitude and methodology that leaves a trail of techniques"

Maybe it's easiest to break it down into its components:

 Neuro - The study of of the nervous system and the mind. How we think, how we experience and how we represent the world to ourselves via our senses

Linguistic - The study of language and communication. The systems through which we represent, code and communicate our representation of the world.

Programming - A sequence of actions. How we motivate ourselves to achieve our goals or react to stimulus
If you stick all that together that covers how we experience the world, how we communicate our experience and how we react to experience, which is just about everything we do really!

Where did NLP come from?
NLP was developed in the early 1970s by John Grinder and Richard Bandler. John Grinder is a world renowned linguist and Richard Bandler studied computer science and then switched to psychology.  They started to study the area of human change and in doing so developed the technique of modelling on which NLP is based.
Modelling is the technique of studying people who are excellent in their field and learning what makes them so good.
In doing so they closely studied Virginia Satir (the "mother" of family therapy) and Milton Erickson (the "father " of modern hypnotherapy) as well as Fritz Perls and others. Virtually a who's who of therapy excellence.

What does NLP do?
NLP enables us to change our behaviour at an unconscious level. 
Look at it this way.  Most people who try to diet find it really difficult to keep up for long. That's because they're consciously trying to behave in a way which is different to the way that their unconscious mind thinks they should.  Using NLP techniques they can change the way that their unconscious wants them to behave and then no conscious effort is needed.
Our behaviours go way beyond over eating though. 
If we hold a deep seated belief that we'll never have enough money our unconscious mind will make sure that we behave in a way which prevents us ever having enough money.  If we change that belief (and we can) to the knowledge that we deserve and can make as much money as we need, then our unconscious will make sure that we behave in a way that makes that happen
If that sounds too simple then consider this:
Most of the sensory input we receive is filtered out by our mind, we simply can't cope with it all.  We make millions of decisions every day (at an unconscious level) about which bits are important and which aren't. What we believe and what goals we're trying to achieve affect all of those decisions.  The conscious decisions that we make about how to act are based on that filtered information, so they're all affected too.  In effect we perceive the world in the way that our unconscious beliefs and decisions dictate to us
The simplest example is deciding to buy a car.  Have you noticed that as soon as you consider a particular model of car you start seeing them all over the place?   Well there aren't suddenly more of them, your unconscious has just decided that they're important so you now notice them. 
Just think what happens when your unconscious decides that great business opportunities are important! or that happiness is important, or finding ways to improve your sporting performance, or finding the right partner... the list is endless

Of course beliefs aren't the only things which drive our behaviour. We have strategies (like computer programmes) for the way we do just about everything. NLP enables us to find ut what they are and, if necessary, change them
We can also learn to largely control our state. Why wonder if you're going to wake up happy and positive tomorrow if you can decide that you're going to?  Have you ever heard the expression  "Doctor Greasepaint"? An actor or performer can be feeling thoroughly miserable and fed up then walk on stage and be full of the joys of spring for the audience and genuinely have a great time. That's controlling your state.