Conflicts within a narrative are usually classified in external and internal conflicts. Both of these types are present in Act I of Hamlet.
In the case of the external conflict, Hamlet is mainly opposed to his mother, and more specifically, to her marriage with King Claudius. After King Hamlet passed away, Hamlet's mother remarried his brother extremely quickly. Hamlet is angry at this union, which he finds immoral and insensitive. The conflict of Hamlet with his mother is an important theme throughout the play.
The second conflict is internal in nature. Hamlet is confused about his role in the new arrangement. On the one hand, he has not inherited the throne, as would be expected. Instead, he feels power has been stolen from him by his uncle. Secondly, as he believed the marriage to be immoral, he feels the need to do something about it. Finally, he also believes his father died in a suspicious way (killed by Claudius) and feels that he has to avenge him. However, Hamlet is indecisive, and full of guilt. He seems incapable of taking any action in any direction, and is overwhelmed by his powerlessness.