The bank and the mailbox are handled differently in GW2. The mail is handled via your UI instead of going to a local with a virtual mailbox. You can mail items to other players but not among your characters.
http://wiki.guildwars2.com/wiki/Mail
Instead, you can use the bank (called the account vault by GW2) to transfer items among your characters which all share the same bank account. To access the bank, talk to a banker or visit a crafting station.
http://wiki.guildwars2.com/wiki/Bank
Note that you can automatically send crafting materials to your bank while out foraging in the wild just by going into inventory and either right clicking your mouse, then selecting "Deposit Collectible" or by clicking the gear icon and choosing "Deposit All Collectibles." This is described in the second link above under the Auto Deposit section.
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