There have been quite a few questions about the Submissions settings and how to view what the students submit.
The first choice you have to make is the Number of Submissions. I recommend that the Unlimited option be used, but that is usually a departmental decision. If you allow unlimited submissions, then no matter how many times a student submits, you get to choose which one you grade. However, if you limit the number of submissions, then someone, somewhere, is going to run out of chances (with a good excuse) and you are going to have to find a way to get a submission from them.
If you do have multiple submissions set, keep in mind that instructors can always see all submissions, so you should look at them all and decide which one you want to take. The Scores interface, by default, is set to only view only the Highest Submission, so you have to click that dropdown menu and select All Submissions. Note that multiple submissions are very different from recording attempts in the recording applet: Instructors only ever see the last recording attempt, but they see all submissions.
The next setting is how you want to deal with late submissions, which most people leave on the default, which is to allow late submissions but mark them as late. The last setting is new for 2008, Auto Submission on the due date. The default for this is not checked, but I would recommend that you check it: somehow it seems logical that assignments that have been saved should be automatically submitted once the due date comes.
So what happens if you do limit the number of submissions, and someone uses up all of their submissions, but has a good excuse and needs to re-take the assignment? Go into the Scores interface and find Submission Status in the menu links near the top of the tool page. Click on that and you will see a list of the students, but below each name, you will see “Allow Retake?” Just click on this link and the student can re-take the assignment – no need to create a new version or anything. If this link is not active, it means that the student has not used up their submissions, and that you are still before the due date. In other words, the student can simply do the assignment again. Students should be aware that each time they start a new submission, the assignment is blank – previous answers do not carry over. Also, they should be aware that the instructor can see every submission, not just the last one.