Can someone save the form they are filling out?

Can someone start to fill out the form and then save it and complete it at a later time?

Yes, you need to enable the Auto Fill feature in your form to allow your users to fill your form later. Check this guide: http://www.jotform.com/help/227-How-to-Enable-Auto-Fill-Feature

Please note that this method will work only when the user is using the same computer and browser to complete the form.

If you have further questions, please let us know.
Thanks.