I am new to that area, so the question may seem strange. However before asking I've read bunch of introductory articles about what are the key points about in machine learning and what are the acting parts of neural networks. Including very useful that one What is machine learning. Basically as I got it - an educated NN is (correct me if it's wrong):
I will assume in my answer that you are working with a simple multi-layer perceptron (MLP), although my answer is applicable to other networks too.
The purpose of 'training' an MLP is to find the correct synaptic weights that minimise the error on the network output.
When a neuron is connected to another neuron, its input is given a weight. The neuron performs a function, such as the weighted sum of all inputs, and then outputs the result.
Once you have trained your network, and found these weights, you can verify the results using a validation set.
If you are happy that your network is performing well, you simply record the weights that you applied to each connection. You can store these weights wherever you like (along with a description of the network structure) and then retrieve them later. There is no need to re-train the network every time you would like to use it.
Hope this helps.