With thousands of food banks on six continents there are many different models. However there are some elements that food banks share.
There is typically one food bank in a city which runs a centralized warehouse. Like a blood bank, that warehouse serves as a single collection and distribution point for food donations. A food bank operates a lot like a for-profit food distributor, but in this case it distributes food to charities not to food retailers.
The largest portion of donated food comes from food leftover from the normal processes of for-profit companies. This food can come from any part of the food chain, e.g. from growers who have produced too much or whose food is not visual appealing enough, from manufacturers who overproduced, or from retailers who over-ordered. Often the product is approaching or past its "expiration", "sell by" or......