Banking in New World comes in the form of a personal storage unit located in each Settlement. You can find the storage locations by opening up the map using the default key of "M" and scrolling in on a settlement until you can see the icons. The storage icon is noted by a box or chest icon on the map.
In order to access your storage just make your way over to that location and use the storage container. This acts just as a bank would in other MMORPGs. You can move items into and out of your storage while accessing a storage container. You can not access your storage anywhere else in the world. You must be standing at a storage container to use it.
The storage capacity is based on weight, not the number of items (just like your character inventory), so you are limited as to what you can store in your storage.
It's important to note that your storage is localized (unless you have connected territories - see below). This means that each settlement has a different storage. For example, you run to the First Light settlement and put 200 ore in that storage. You then make your way up to Everfall. While in Everfall you use the storage container in that settlement. The 200 ore you stored in First Light willnot be there because it's in the First Light storage and not the Everfall storage. So, the only way to get that 200 ore would be to then go back to First Light and retrieve it from the First Light storage.
Connected Storages via Connected Territories
Any two settlements controlled by the same faction will have their settlement storages linked. Players can transfer items from a storage linked to their current location for a coin fee. If the territories are not linked then your storage is localized as stated above.
You can increase your storage capacity in each settlement by increasing your standing within that territory and choosing the increase storage capacity bonus. When you increase you capacity in one territory it applies only to that territory. It is not a global increase for every storage location.