In simple words: Batch size is examples per update; an epoch is one pass through all training examples.
One full pass
An epoch means the model has processed the whole training dataset once. With 1,000 examples and a batch size of 100, there are 10 batches in one epoch.
How many epochs
The team chooses a maximum and checks validation results while training. Too few epochs may leave the model undertrained. Too many can make it fit old examples too closely. Early stopping ends training when validation performance stops improving.
See the flow
examples→batches→one epoch→validation check
Simple example
A student studies all 1,000 practice questions once: one epoch. Ten groups of 100 questions are ten batches.
Remember: Batch size is examples per update; an epoch is one pass through all training examples.