How is the learning process for a NN implemented? Thanks to BP and some scoring/loss functions, a NN can train to do the job as demanded. Or at least try to do so.
I think I've understood BP but still I wonder about the following:
Let's say there is a certain NN and the input is a single number. The NN will propagate the information through all layers and all weights and activation functions and so on will finally output a single number as well. If the output number is the same as the input number, nothing will happen. In case, it is different, the error will result in a change of the weights such that the NN will provide this number according to that certain input number.
I think I also understood this.
What I don't get is: How is it done such that the NN can handle multiple/different inputs?
Assume $x = 3$.
The NN will now adjust the weights to provide this as output.
Next iteration, $x = 8$.
The NN will now adjust the weights to provide this as output.
From my understanding, the NN is now trained to put out $8$. And it lost its capability to reconstruct $3$.
Obviously this is not the fact. So how is it done that the NN can handle both the $x = 3$ and $x = 8$?
Something like a lookup table is probably too plain (and dumb). I think the NN learns to generate something like a (single) function. Does that make sense?

Despite that, what does the implementation concept look like? Just the idea of it. 
 A: A node is a (usually non-linear) function that combines the input. The entirety of the neural network is an additive model of nodes that in turn are connected to either the next hidden layer, or the output layer. Training learns the weight of connections: How different combinations of the input relate to the output.

From my understanding, the NN is now trained to put out 8.

This is where your confusion arises. The networks has learned neither to output $8$, nor $3$, but instead how to combine the input such that the predicted output is close to the actual recorded output.$^\dagger$ 
I may be reading too much into your question, but it also seems you might be confused about what the input is: It is possible for the input to be the same as the output (this is the case for autoencoders), but more generally, you have a completely different input (e.g. images of animals) used to learn the output (e.g. the type of animal).
$\dagger$: I say close because we do not want to learn how the input relates output of the sample, but rather how the input of any sample from this population relates to the outcome of any sample from this population. If it would predict exactly the recorded output in the sample (e.g. $8$ and $3$), the problem is either very simple, or it is overfitting.
