A long term girlfriend/boyfriend, or your wife/husband. So called because they are very special to you and it is as though they complete you.
Me and my other half went on a night boat trip in Paris last weekend, it was lovely and romantic...
Other Half is what you call the person in your life that is so much like you and you spend a lot of time with.
Person 1: Why are you so distant?
Person 2: My Other Half isn't here
Person 1: I'm so Sorry:'(