The stash is the large chest that will appear in each of the quest hubs found throughout the game, and they are used to store items that you don't want to have in your main inventory, and can also be used to exchange items between your normal mode characters (which is the stash I recommend you upgrade, unless you fancy Hardcore).
If you don't have the cash to purchase upgrades that you need early on, creating a new character (or characters) and moving items through the stash to have the items be stored in their inventory is a free way to store tons of stuff early on. With the vastly increased drop rate for powerful items, having as many slots available as possible will become useful earlier.
The upgrades are as follows for a total cost of 1,020,000 gold
10k from 20 to 40 spaces
10k from 40 to 60 spaces
25k from 60 to 80 spaces
25k from 80 to 100 spaces
75k from 100 to 120 spaces
75k from 120 to 140 spaces
150k from 140 to 160 spaces
150k from 160 to 180 spaces
500k from 180 to a max of 210 spaces
Thanks to MrJMcWild for the price list.