The relationship between these two started when Izaya told Saki to meet with Masaomi. When she finally approached him, Masaomi felt a strange sense of mystique coming off of her and was immediately intrigued. The two of them decided to continue to see each other and eventually, genuinely, fell in love. The first major turning point in their relationship would occur during the gang war between the yellow scarves and the blue squares which culminated in Saki being kidnapped and Masaomi being too afraid to save her on his own. For almost a year after this incident, Masaomi would be too ashamed to face Saki in person at the hospital. Their mutual feelings for each other remained strong despite their time apart as revealed towards the end of the Yellow Scarves arc with the two of them leaving town to start a new life together.
Masaomi cares for Saki very deeply and the day he failed to save her was a source of great guilt and self-loathing throughout the early portions of the series. As stated by Izaya, this guilt is what would cause Saki to become a sort of "god" in the eyes of Masaomi in the sense that his failure to save her would become one of the primary driving forces behind his actions from that day onward. This prediction came true although it did eventually result in a positive outcome for the both of them.
Saki, although initially she stayed with Masaomi because it was what Izaya had told her to do, she eventually fell for him on her own as they spent time together to the point that she would even be willing to sacrifice herself to the Blue squares if it meant they would end the war with the yellow scarves. She proves her love again when she betrays Izaya, a man she practically worshiped, in order to save Masaomi from getting killed by Horada.
The first real conflict to emerge in their relationship was Saki's reverence for Izaya which Masaomi wanted to free her from when they first started dating. While she cares for Masaomi more than Izaya, it is clear that she still holds some sort of respect for him which annoys Masaomi to a certain extent.
Another aspect of their relationship is explored during the Akane Arc where Masaomi still feels a sense of loneliness despite living with Saki. She realizes that while she can be his lover, she can't be a "friend" to him in the same sense that Mikado or Anri can and encourages him to repair the ties he had to his friends instead of trying to force himself to be happy.
Finally, although trivial compared to the previously addressed points, another conflict occurs in their reliance on technology. Masaomi frequently relies on social networks and forums in order to stay connected with people where as Saki almost never uses them (although she did join the chat room group at the behest of Masaomi) unless absolutely necessary.