When you start playing Stardew Valley, you only have a single row of inventory space. This row consists of only 12 slots, so you can't really pick up a lot of stuff, but it's possible to upgrade it to 2 rows with 24 slots, or even 3 rows with a total of 36 slots!
In order to upgrade your inventory size, head to Pierre's Shop and click on the backpack that lies on his desk. This will give you the ability to purchase an inventory upgrade.
Upgrading to 2 rows will cost you 2,000g.
Once you have the 2 row upgrade, another upgrade into getting all 3 rows will cost you 10,000g.