Stash is an addon to track, browse and search for items stored across all your characters.

As you play, Stash performs a low-impact monitoring of your items as they move to/from your worn gear, bags, bank and mail. On modern clients it will also track void storage, reagents, warband and currency.

When you need to know if one of your alts has Wool Cloth for your new tailor or any other item you've accumulated, you can summon the window in one of several ways:

/stash command
Key binding defined in the game's key binding interface
Minimap button if you enable it in options
A Broker addon if one is enabled

The addon should be self-explanatory how to use:

The resizable window has a search box to search for items.
In the search results, mouseover an item in the list to see who has it stashed and where.
A button in the topleft opens a pullout panel for filtering to a specific character or a specific type of item.
A button in the topright opens settings to change some preferences.
The total gold across all characters is displayed at the bottom of the window.
Clicking the total gold will display a breakdown of gold on all your characters.

(Users of Possessions will find many similarities. Stash was created to replace Possessions for my personal use; but others may find it useful too.)

Notes:

To minmize time spent scrutinizing items, items are tracked by their itemID without customizations. This means 6/6 Korthian armor will appear as their base versions, mithril spurs won't appear on boots, and Plate Helm of the Eagle will list as just Plate Helm without the suffix.
On modern clients, caged battle pets are also stripped down to their speciesID. This means their stats (including rarity) are not available.
Stash is a traditional addon that can only observe what it sees in game. It will only track items observed on characters who have logged in while Stash was enabled. If you receive mail while away from a mailbox, it can't know what you just received. If you never visited the bank on an alt, Stash can't know what's in the bank on that alt. If you never logged in an alt since installing Stash, the addon doesn't know the alt exists.
Stash is not intended to be a tooltip addon.