Older mobile phones would store received text messages on your SIM card rather than in their own (very limited) memory. Service messages sent to you by your mobile provider are also stored on the SIM card, even today.
Modern phones store inbound text messages in their own (significantly more capacious than their predecessors') memory or even, in some cases, on a removable memory card (miniSD or microSD most of the time). Nokia phones based on S60v3 with feature pack 2 and later (N78, N96, 5800 XpressMusic, N97 etc.) detect messages on the SIM card and, if they find any, alert the user to their presence by displaying a flashing envelope on the screen.
The only way to get rid of it is to delete the messages on the SIM card. To achieve this:
Menu > Messaging > Options > SIM messages
Select all the messages visible here:
Options > Mark/unmark > Mark all
Then delete them by pressing the "C" key for phones that have it (keypad-operated phones) or by tapping "Options" and selecting "Delete" for touchscreen-driven phones.
The messages on the SIM will be deleted and the flashing envelope will be removed.