Placing Items

You can place items into a bag by left-clicking and dragging them over to the bag you want to place them in. This will place the item in that specific bag. This allows you to open up your backpack (or other bag) and quickly move items out of and into other bags. If you right-click while moving an item, it will be returned to its original location (i.e. won't be moved).
When you right-click on the loot dropped from monsters, the items will automatically be placed in one of your bags.

Additional Bag Slots

Players have four additional bag slots to the left of the backpack slot.
You can find, trade, or buy additional packs to place in these extra bag locations. The bigger the bag, the better it is. The first bags you are likely to find are 6-slot bags.

You can open up all your bags at once by hitting "Shift + B." Some players remap the B hot key to open up all their bags at once.

Removing Bags from Bank Slots

To remove a bag from one of your additional bag slots you must first empty the bag. You can put the items into another bag, or the bank, or destroy them. Once the bag is completely empty, you will now be able to pick it up by clicking on that bag's icon and dragging it. You can now place it into another one of your bags, sell it, trade it, or destroy it.

Dropping Items

You cannot drop items on the ground, so if you want to get rid of an item, you must drag it outside of your bag and then left-click. You will receive a pop-up box that asks you if you want to destroy that item. Make sure you want to destroy it before selecting "Yes".

Bank

The bank sells bonus slots (think of them as bag slots) that you can buy to place fully loaded bags in. You can buy a bank slot and then use it to store a bag full of Copper Bars for later, or whatever else you want!
Save any extra bags you find to use at the bank. Obviously, the bigger the bag, the better it will be to use for this. Normal slots in the bank (free ones) cannot hold loaded bags. They can only hold empty bags.

When a player places a bag into a purchased bank bag slot, it becomes a soulbound item.

Ammo Bags/Quivers

A "Ammo Slot" is located on the character screen. Ammo must be placed in this slot in order to use a ranged weapon (bow, gun, crossbow). Only ammo placed in this slot, and any like it in your bags, will be used. Other types of ammo will not be used.

Ammo will be fired from a quiver, ammo pouch, or any other bag.

When you right-click ammo in your bag/quiver, it will place the ammo in the "Ammo Slot".

You can also hotkey ammo in your action bar to quick load.

Other information

Bags can have special magical bonuses on them in addition to being able to carry items.

Monsters and chests sometimes have bags.

Tailors are able to make bags. Save up scraps of linen, bolts of cloth, and wool to give to a tailor to make bags.

Bags are sometimes given as quest rewards.

Tips

Look for bag stores in big cities.

Get different colored bags and designate them for specific things. You can put all your quest items in a red bag, and trade skill items in a blue bag, and all the items you want to sell in a green bag. This allows you to be more organized!