The drama of Downton Abbey begins when the Earl and Countess of Grantham's heir to their family estate (and eldest daughter Mary's marriage prospect) is killed in the sinking of the Titanic. Since Mary is one of three daughters, and the heir has to be male, the only way she can now inherit the estate is by marrying her cousin Matthew.
As was the case for powerless women in the Edwardian period, Mary is under an immense amount of pressure to get married to secure her position and fortune. In 1912, the marriage rate in the UK for women was twice what it is today. And at 20 years old at the start of the series, she's practically a spinster of the times, since she's already had several "seasons" to find a husband.