Achievements and achievement badges provide motivation and progress tracking. Badges and associated points are awarded for completing tasks on the site. These tasks include achieving personal bests, achieving volume based goals, such as number of problems solved or games played, success and activity streaks and exploration of site features.
Your preferences has a tab for achievement related settings, and if you would like to turn off achievement notifications you can do so there. Achievements will still be tracked, but you will need to go to your achievement stats page to see which achievements have been completed. If you would like less obtrusive noticiations, change notification method from dialog to notification panel which will use an automatically hidden slide out panel to the right of the page to display all achievements rather than a large dialog centered in the page. Note that for frequently awarded achievements they will often be shown as notification panels instead of dialogs to avoid disrupting your sessions.