Mary: Lady Mary Josephine Crawley is the eldest daughter of Lord and Countess Grantham. Mary was informally engaged to her cousin Patrick Gordon, the Lord's heir presumptive, but he died in episode one, throwing the plans for the estate into question. Much of season one revolves around trying to find Mary a new suitable husband, but Mary slowly becomes less and less interested in just being married off, rebelling at one point by giving her virginity to a Turkish dignitary who then dies mid-coitus. Mary is a somewhat infuriating character; she has gumption, but she's also caught between the old-fashioned way of doing things and the modern and more progressive way the world is headed, causing her to make some boneheaded decisions (like not agreeing to marry Matthew right away!).
Matthew Crawley: The heir presumptive to the Crawley estate, Matthew, a lawyer, is Lord Grantham's third cousin once removed who moves to Yorkshire with his mother in order to become more familiar with what will eventually be his. Matthew initially was considered as possible husband for Mary, as their marriage would ensure that both her title and the family money would stay with Lord and Countess Grantham's lineage. While Mary was not initially taken with Matthew, they developed a bond and by the end of season one, Matthew had proposed, but Mary took to long in deciding whether, despite being in love with Matthew, she wanted to marry him. Because Cora was pregnant at the time, Matthew took this as a sign that Mary was only interested in marrying him if she was sure he would be the heir and her mother's pregnancy (which eventually miscarried) made that unclear. Matthew remained the heir presumptive at the end of season one and Mary was left heartbroken.