The first most obvious conflict is social conflict. People are judged based on wealth and social status and even marriage is determined by one's social ranking and wealth. Love is secondary. It can even be argued that the other conflicts in this story are all related or a result of this main conflict.
The conflicts that exemplify this social conflict are:
Marianne vs. Willoughby
Elinor vs. Lucy Steele
Edward vs. his mother
Wiloughby vs his aunt
Each of these above conflicts is stemmed from wealth and social status as a barrier to true love. Willoughby has a conflict with Marianne because he cannot marry her since she does not have enough money. He has a conflict with his aunt because he needs to act to her wishes because he needs her money. Elinor and Lucy have a conflict over who will get Edward and who has a greater stature with his mother to be approved for marriage. Edward has a conflict with his mother who will disown him if he does not marry the appropriate social/wealthy woman. All of these conflicts deal with marriage but not because of love, but rather the conflicts are based on money.
There is internal conflict which plagues Elinor throughout the novel. She struggles with hiding her own feelings and her opwn pain to protect her family - specifically her mother and sister. She constantly struggles with what she should keep to herself and when she should become involved in her sister's life.
Several conflicts exist in the novel: wealth versus poverty, passion versus reasons, marrying for love versus marrying for security. The individual conflicts between characters exemplify these thematic conflicts.
There is conflict between Elinor and Marianne Dashwood. Elinor represents the sensible, reasonable nature, Marianne the romantic, passionate nature.
Marianne is in conflict with the Colonel Brandon who wants to marry her. However, she believes he is too old and sensible for her.
Marianne also has a conflict with John Willoughby with whom she is in passionate love but by whom she is later rejected.
Willougby also has a conflict with Elinor over the manner in which he rejects and hurts her sister. Later it is revealed that Willoughby must marry for money in order to secure his place in society, so he has not rejected Marianne for love but for financial reasons.
Elinor briefly allows herself to abandon reason and fall in love with Edward Ferrars. When she is rejected, she internalizes the conflict and chastizes herself for abandoning her sensibility. Edward, eventually, decides to follow his heart rather than his mother's wishes in finding an "appropriate" mate and returns to Elinor.