Apologies if you already know this, but...
The check mark indicates that the message was sent (ie. transferred from the handheld to the BES) successfully.
By default the messages list shows sent and received items.
If what they actually want is to not see sent mail, then depending on the model, ALT-I should work. It applies a display filter to show only incoming messages.
ALT-O shows outgoing messages.
There's a few other posts on this subject - do a search for 'sent items', but click here for one