Mining is the glue that holds ethereum's 'decentralized app store' together by ensuring that it comes to consensus on each change to any of the applications (dapps) running on the network.
Miners set computers loose to solve cryptographic puzzles in an attempt to win ether, and they need to try a huge number of computational problems until one unlocks a new batch of the asset.
The catch is that mining on major public blockchains tends to require more and more power over time, as more people invest in more powerful hardware.
In this day and age, those mining with low-powered setups are unlikely to win, but it's still a viable past-time for hobbyists and enthusiasts alike.