Currently, the bank functions similarly to a large chest. Items stack in the same way as the Inventory, and has the ability to get full. The contents of the bank do not delete on Death, so it is a perfect place to store items you do not immediately require.
An update to the bank is in the works, to make stacks infinite (similar to the Grand Exchange) and to make it paginated (allowing for more types of items, and grouping).
When a player does the quests on Tutorial Island, there is a part where they HAVE TO open the enderchest. Otherwise, the feature of their bank will not work. If, for some reason, you are unable to do this, there is one enderchest in the Lumbridge castle dungeon. It can be found threw a hatch in the kitchen.