Are you playing in your browser? Depending on your settings, your browser may automatically clear your cookies, and in doing so delete your saved game. If you download the game ROM you can play in your Game Boy environment of choice, and have direct access to your saved games. Regardless, check your browser settings regarding temporary files and cookies if you intend on playing GB games in a web browser
Viewing post in Taiyaki Fabulous Museum of Fish comments
There are a number of Game Boy emulators that you can use, many of which are free and open source. While I personally use Bizhawk, something like Visual Boy Advance might be easier to get started with.