A non-exclusive relationship is casual dating where two people connect emotionally, and sometimes physically, but there is no commitment. This is not a serious relationship, so you are free to see other people.

What is a committed relationship called?

A committed relationship is an interpersonal relationship based upon agreed-upon commitment to one another involving love, trust, honesty, openness, or some other behavior. Forms of committed relationships include close friendship, long-term relationships, engagement, marriage, and civil unions.