This is a great question and I would say that there are three main conflicts in the book.
First, the most obvious conflict is based on class. Noah is a great man, but he does not come from money and privilege. This is a point of contention, because Allison does come from money. Moreover, her mother is very keen on keeping Allison from Noah. She want to protect her by allowing her to marry within their social class. This socio-economic barrier is one of the strongest in the work.
Second, there is also a family conflict. As was intimated above Allison's mother is only trying to look at for her daughter. However, she play a huge role, because she does all in her power to keep Noah and Allison at arms length; she almost succeeds.
Finally, there is a biological or physical conflict. In Allison's later years, she is not able to remember Noah and their life together. So, Noah tries his best to win her over every time with the reading of a notebook.