Note: This description comes directly from my brain. I learned about Neural Nets a very long time ago, so I cannot be sure of the accuracy. [if you have better info, please feel free to send it to me, you can reach me through the 'Contact Me' link on the top of the page - Thanks ]
A Neural Net is a construct that takes an input and makes an output. It learns what the correct outputs are by being 'shown' an input and then 'told' what the correct output for that input is, I call this a lesson. After enough lessons it learns to provide the correct output when given an input. Both the input and output are numbers.
An example is a Neural Network that is connected to a camera produces a grid of 64 pixels that are either 1 or 0 (think of it as a very low resoultion black and white camera). The 64 pixels are inputs to the network. The output of this network is 2 bits, 'is it a car' and 'is it big'. To train this network it is 'shown' pictures of cars (that is the numbers at the inputs are the numbers that the camera's pixel grid makes when a picture is presented to it). When it is shown a picture, its outputs are set to the proper values and it gets trained. After it has been trained, it can be presented with an input and it can say if it 'sees' a car, and if that car is 'big' or not. Note that you could teach it in such a way that the big/small bit is random when it is not a car, or you can teach it to have a certain big/small value when it is not seeing a car