FogBugz alerts you via email when a case is assigned to you, or when a case assigned to you is edited by someone else.
Email notifications can be turned on or off by navigating to My Settings > Options.
Any time a case is assigned to you, or someone edits a case currently assigned to you, FogBugz generates an email notification. To keep from cluttering your inbox with redundant notifications, FogBugz does not notify you on actions you perform yourself.
Note that this "no self-notification" also applies if you don't directly edit the case in FogBugz, but instead send an email into FogBugz. If the From: address on an incoming email matches the user who has the case assigned to him/her, that user does not get notified.
You can also set up subscriptions to receive notifications for different types of cases, including cases that are not assigned to you. To learn more, check out this article.
Because filtering capability is so much more robust in email clients than it likely ever will be in FogBugz, we tend toward the more-is-more approach for notifications. These notifications are predictably structured, so they can be easily filtered by Outlook, Thunderbird, or Gmail. For instance, you can filter out all emails with A case which you opened has been resolved in the body and Inbox in the subject line.
If you're not getting email notifications, you can find a troubleshooting guide here.