You can find a good summary of Twelfth Night in the enote on the play. (See link below.)
As far as their relationship, it changes a bit depending on the situation, but essentially it is one of seeking love and missing due to surface misunderstandings until they finally find one another. To be more specific, Viola disguises herself as a man while in this foreign land and enters Orsino's service. While disguised she falls for Orsino, but he's in love with someone else. She carries his message of love to this woman (Olivia), all the while getting to know him better. At the same time, Orsino gets to know Viola in disguise, appreciating her good qualities. In the end, Viola is revealed as a woman, and the two can be together in love.