The manga you're looking for is Maison Ikkoku.
The story takes place in Maison Ikkoku, a worn and aging boarding house where Godai Yuusaku, a 20 year old college applicant, lives. Though honest and good-natured, he is weak willed and often taken advantage of by the offbeat and mischievous tenants who live with him. As he is about to move out, he is stopped at the door by the young and beautiful Otonashi Kyouko, who announces she will be taking over as landlord. Godai immediately falls in love with her and decides to stay.
Later, Godai and the other tenants find out that despite her young age, she is a widow, as she married her high school teacher, but he died shortly after their marriage. Godai empathizes with her and endeavors to free her from her sadness.
He manages to work up enough courage to confess his love to her, and it begins to look like that a relationship between them might actually appear... until Kyouko meets the rich, handsome, and charming Mitaka Shun. Mitaka quickly declares his intention to court Kyouko and states that he is very patient and can wait until her heart is ready.