HESI is used by some but not all schools, as a way of seeing if students are likely to pass the NCLEX. Some schools will not let you graduate if you don't pass the HESI. Others use it to see where their students are academically, but don't tie graduation to it.
The NCLEX is the exam, sometimes known as "taking your boards", used nationally by the state Boards of Nursing, who regulate nursing for their individual states, to determine if they will register you as a nurse, qualified to practice nursing in that state. Its the National Council Licensing Exam for Registered or Practical Nurses.