Players start with a permanent 16-slot backpack. Press the "B" key to open your backpack, or click on the backpack in the bottom right of the screen. You can now move items in or out of the backpack. The default backpack cannot be moved or sold.

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.