To close a task, click on the current status on the top right corner to expand a drop down. Select Completed from the dropdown. If the option to complete a task is not available, the task may be a parent to open subtasks. Click on the Subtasks tab to view the subtasks. All subtasks must be completed before the parent task will have the option to complete.
Users will need to have view and edit permissions in the tasks module to perform this action. Depending on the user's project permission settings, they may be able to view all project tasks or just a subset. Watchers specifically added to the task will be able to view it.