What happens after backpropagation in the learning phase within Neural Network? Within a Neural Network during the learning phase ,there are two mechanisms happening. Feedforward and Backpropagation. Taking an example of XOR operation.
A   B   Q
0   0   0
0   1   1
1   0   1
1   1   0

For the first pass(0,0->0) feedforward takes place, then backpropagation happens.After this step all the weights are recalculated .
What happens now? 
Question-1:Again the same input 0,0 is feedforward with the new calculated weights(during the backpropagation) and then backprogated until the error becomes nill? If yes, what happens if the error is never nill? Which brings me to the next question.
Question-2:When would the learning for the next pass(0,1->1) happen?
 Question-3: Suppose the concluded weights for the 1st pass is 10.3,-2.3,5.5 .The second pass(0,1->1) starts the feedforward with the concluded weights of the first pass?
 A: The answer depends on which algorithm you are using. For instance, a naive algorithm would be to calculate the loss on entire training data set (sample), then get the numerical gradient. Then use the gradient to suggest the step change in parameters. Changing parameters is learning, as long as your loss is decreasing. Hence, each change in parameters requires you feed forward entire sample.
You could go another extreme and change the parameters at every observations, i.e. batch size = 1. You feed forward just one observation, and this lets you estimate the gradient. The gradient will be very noisy, of course.
Somewhere in between is the batch size >1 but less than sample size. Here, you could smooth out the noisy gradient estimation by running a few observations before adjusting the parameters. It turns out that batch optimization is most optimal usually.
A: No Same inputs is not passed, you iterate over each instance in training set one by one till you iterate all instances in training set(this you do as many times as your number of epochs or iterations that you provide as hyper parameter). 
While iterating over each you adjust weights as per each instance, Over multiple iterations. 
As you have four instances and lets say you ran it for 50 iterations. 
So yes each instance will be sent to the Network 50 times, but not all 50   together.
All instances(training) will be sent one by one 50 times.
Yes, for question 3. but first pass doesn't mean 50 instances of 0,0. it just mean 1 instance of 0,0 then over that we pass 1,0 and so on fifty times.  
Hope this answers multiple questions that you have.
