When you go exploring in the mines, you will eventually end up finding geodes. There are currently four different geodes in the game: Geode, Frozen Geode, Magma Geode, and Omni Geode. They all have different loot tables, and the ones you tend to find lower in the Mine has generally better loot.
Before you can do anything with the geodes you will need to process them, which basically means that you pay to open them. To process geodes in Stardew Valley, head to Clint the Blacksmith. He will give you the option to pay 25g for processing a single geode, and you get to keep the loot inside it.
Clint will generally offer to process geodes for you when he is in the shop, unless he is currently working on upgrading your tools. If this is the case, return later when he is done with upgrading your tools.
Even tough it cost some money to process the geodes, it generally gives you a great return of profit, since most of the geodes has a chance to contain minerals that are worth quite a lot more than the 25g fee. So it is almost always worth spending g to process all the geodes you find, especially early in the game when you will need money.