A checkmark after a trade partner’s name indicates that the partner has been confirmed for that task, meaning that he or she has committed to performing that task on that project on those dates. Trade Partners are confirmed either by the partner clicking on the confirmation link in their notification e-mails (if their profiles are set to send them notification e-mails) or someone on your team has checked it off.
Click the icon on the schedule (in the "Notifications" column of the Gantt or Task View) to see who confirmed the partner and when.