Something that's open-ended doesn't have a specific end date or time. You might buy an open-ended plane ticket to Rome if you're not sure when — or if — you'll want to come home.
Things that are limitless, or that have no restrictions or restraints, can be called open-ended. An open-ended lease on a apartment doesn't end on a specific day, and an open-ended babysitting job would be somewhat casual, rather than a regular, reliable job. Open-ended questions are complicated ones — they can't be answered with a simple "yes" or "no." The adjective open-ended has been in use since the mid-1800s.